Simple Way to Master Indices Maths Question

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Have you seen many mistakes like the below?


Question 1:
3^x = 27

Solution:
3^x = 3⁴ ===> x = 4

Question 2:
9^x = 3³

Solution:
3^3x = 3³ ===> 3x = 3 ===> x = 1

Why the error?

A simple explanation is that the maths learner is not familiar with the basic multiplication of repeated numbers.

3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49

2 x 2 x 2 = 8
3 x 3 x 3 = 27

2 x 2 x 2 x 2 = 16

etc.....

Once you have mastered this basic repeated multiplications, you can rest assure that indices question will not be there to haunt you.

How about solving "x" in this 9^{x + 1} + 2(3^x) - 3 = 0 ?

I bet that if you understood the above criteria of learning indices, the equation can be easily solved for x (using quadratic formula as a hint).

All complex things start off with simple things.
Do you agree this applies to maths?

:D

Indices | Interesting Mistakes (3)

If you are not careful with the base and power in a mathematics index expression, simple mistakes will occur. These simple mistakes will expose the weakness in your basic understanding of indices concepts and principles.

But, although, mistakes do occur, rest assure that learning from them is a good thing to have in the learning process.

What is the common mistake student normally make when doing indices?

Here is an example:

y = 9^{x + 1}

y = (3²)^{x + 1}

y = 3^{2x + 1}

You see the error here?

Yes, the power to the base 3 is wrongly done.

It should be 2(x + 1) = 2x + 2. The last term was left out of the multiplication.

A good way to prevent this mistake, or slip-of-the-mind error, is to use parentheses.

Using parentheses at the (x + 1) index will visually group up the "target" for multiplication by 2.

The correct answer: y = 3^{2x + 2}

Take note of this simple maths error and you are on the way to a happy maths learning journey.

Cheers! Making maths interesting goes a long way.....

:-)

Indices | Interesting Mistakes (2)

For maths problem related to indices, you do not only look at the power. You have to take care of the base too.

The common mistake is to ignore the sign of the base when doing computation involving index.

A simple example illustrates the error that is very common when learning maths.

Example:
Solve 3x² - 4x + 1 = 0 using the quadratic formula.

Solution:
In solving, we need to extract out the a = 3, b= -4 and c = 1 to fill into the quadratic formula.

However, the quadratic formula requires the utilisation of the b² - 4ac expression.

Here the mistake is to fill in b² as -4² = - 16!

This shouldn't be the case. It should be (-4)² = 16.

It is a difference of the sign (positive versus negative).

Thus to handle question on indices, you will have to be extra careful on the base and its sign. Make full use of parentheses, if possible, to group and focus the targeted term or number.

In summary, index or power affects whatever it includes, base and its sign together.

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U

Indices | Interesting Mistakes (1)

I came across an amazing mistake on the index conversion which made me thinking.

One student wrote: a^-1/2 = a²

What is the going in his brain?

He is not completely wrong. He had applied something related to the properties of indices. But had confused the application through improper usage.

This may be the result of trying to memorise the working instead of fully understanding the mathematics principles.

I suspected he may be using the idea of a^-m = 1/a^m.

However, instead of changing or converting the numerator as a whole, he converted the index only.

Though the mistake made was minor, it created something of a surprise. Many interpretation came out of the simple index law.

Although formula is given at times to aid the solving of mathematics questions, the correct interpretation and understanding of the concepts and writing of the terms has to be digested with clarity.

Otherwise outcomes may appear that amuses the teachers.

:) Maths is fun when looked at from some angle.

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Division of x^n with y^n

Confusion does happen when you are bombarded with many numbers, exponential and its likes.

After dealing with indices, logarithm and their multiplication and division, the brain will sort of tangle up and produces weird happenings.

Take 2 examples below:

1) x^n / y^n ==> x /y

2) log x^n / log y^n ==> (log x) / (log y)

By looking at the first example, you may find nothing wrong.
Since the power "n" is similar for the numerator and denominator, you can do the normal cancellation as you do for "mx / my" = x/y.

However, something may tell you that something is amiss.

While "mx / my" is truely x / y, this is because mx means m times x.
There are "m" number of x that are ADDED up.

For x^n, it means "x" is multiplied by itself n times. (or x times x times x times x ....)

Thus x^n is not equal to xn.
The truth of "cancellation" is that since a / a = 1, and this "1" is not required to be written, the disappearance seems to be "cancellation".

Let me explain further with an example(A).
ax / ay = (a/a)(x/y) = (1) (x / y) = x / y. The "1" disappeared and seems to be cancelled off.

The mistake made in Example 1 in the beginning, is the assumption that the powers "n" followed the concept of "ax" in example(A).

Correct answer for x^n / y^n = (x/y)^n. ==> The powers of n are not removed.

Now for the Example 2, at the beginning, it showed a similar cancellation of the powers "n".
But this time round, it can be said to be conditionally correct.

Why?

If the idea that similar "letter" of "n" in the power can be removed through cancellation, then the answer, although correct, is theoretically wrong.

However, if you know that using the Power Law of logarithm, log x^n can become "nlogx" and therefore, log y^n can also be "nlog y", the result of (log x) /(log y) can be rightfully considered correct, since the "n" is removed according to the idea that n/n = 1 and disappeared, or qualified for removal.

In summary, mistakes do happen when the concept of power (x^n) and pure multiplication (x times n) is not clearly understood.

Cancellation of "letters" or symbols in math expressions should be highlighted as a shortcut to removal due to being "1" that can be omitted in the written form.

This concept of "cancellation" is easy if you understand that it is because of a/a = 1.

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Negative Value Equated to Exponential Expression

This is a very interesting maths question, that may trick many students.

2^x = -1

What is the value for x?

I got all sort of answers.
Some gave me x = 0 (knowing that 1 = anything to the power of 0).
Some treat the expression to be 2x = -1 ==> x = -1/2.
Some used the calculator, applying "logarithm" operation and getting an "Error" message!

Those who know the answer, congratulation. You can stop reading this post.

For those interested and wanting to know what's the answer, read on...

Knowing the answer, is fine here. But knowing with understanding is better.

Now, let's put some numerical values of x into the expression.

If we choose x = +3, y = 2³ = 8 ( a positive number).
If we choose x = -3, y = 2^-3 = 1/8 ( a positive number).
If we choose x = 0, y = 1 (again a positive number).

Conclusion: All the x values substituted will get us positive number instead of the desired negative number.

Then how do we get a NEGATIVE number from the exponential expression?

The answer is:
WE can NEVER get a valid numerical value for x for expression in the form a^x or equivalent when it is equated to a negative value.

Clear?
Do not fall into this mathematical trap. You will not look good, if you can solve it!

Have fun testing your classmate.

:-)

Solving Index expression in Quadratic form

In maths, the solving of indices questions and its quadratic counterparts are a common practice.

Depending on how you approach the solving, you may encounter a tough journey or a smooth-flowing one.

There are, however, simple tools and concepts that can be applied, in order to have fun solving them.

Here you go....

First, let's us take an example.

Example:
(2^x)² + 3(2^x) - 4 = 0

To solve this type of quadratic (index) equation, you have to take note of the common mistake in mis-interpreting the second term above.

This is the "3(2^x)" term. Refer to this link for explanation on the mistake.

Next, apply the concept of using "let" to the given equation.

This is needed to simplify the mathematical expression visually. Otherwise, it may look intimidating.

To learn about the power of using "Let", click here.

With the above 2 basic steps adhered to, you are ready to move forward into a relax solving environment to handle the given equation with ease.

Simplified equation: (After letting y = 2^x)

==> y² +3y -4 = 0

Applying next, the quadratic formula method, you can see that a = 1, b= 3 and c = -4.

Solving it for y, you will get 2 values shown below. ( Click here to learn how to make use of quadractic formula to solve.)

==> y = (-3 + 5)/2 = 1 and y = (-3-5)/2 = -4

After which, solve for x.
This y is related to x by the "letting" operation you have did in the first place, that is, y = 2^x.

y = 1: y = 2^x ==> 2^x = 1 = 2⁰ ==> x = 0 (Answer), logical comparison.

The other answer of y = -4 will not yield any valid real answer for x here.
( Why? --- see my next post).

So, you have done the solution very easily and without hiccups if you have understood the basic concept. If you have reviewed the working here, you will notice that there is nothing complex with all the steps.

Maths can be solved through a series of mind-blowing steps. But the reverse can also be true. It is up to you to define and choose the desire path.

Do not despair initially, as you need experience to manage this selection of strategy. How to achieve this experience? Simply practice and practice.

:-) ***

Logical Comparison in Maths

Given a maths question, most of us will attempt solving it step by step, diligently, using formula and methods we have mastered.

However, there are times when a simpler solution can be done if we are able to see the logical side to the maths problem assigned.

Let me show an example.

Question:
2^x = 2⁴, find the value of x.

Solution: (mathematically)
Taking "log" on both sides, ==> x log 2 = 4 log 2 ==> x = 4 (log 2) / (log 2) = 4

Solution: (logically)
Comparing the values of their power, we get x = 4, since their base is the same (=2).
No working is needed!

Thus, maths does not actually just train us to do things systematically, it allows us to have a bit of mental freedom. This freedom is done in terms of the small little "twists" that make use of visual comparison or logical thinking (comparison).

Interesting approach to maths learning, right?
So many ways, mathematically and non-mathematically.
But best is the stretch to our mind to develop it to see things in many angles.

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Tricky Square Roots

The simplified expression after the square rooting the top expression seemed easy enough.

The individual terms within the square root are handled separately to obtain what is shown.

It seems to be correct and neat. Or is it?

This is a very common mistake made by many students.

It is not correct. Do take note.

Why?

The principle of indices do not permit this type of computing.

Addition of terms has to be taken as a whole (after summation or as a complete group).

The square root is actually to the power of (1/2), which is similar to that of x².

(x + y)(x + y) = (x + y)²

[(x + y)(x+ y)]^1/2 = (x + y)

Think about it.

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Ways To Improve Logical Learning Part Of Maths

In learning of maths, as with learning of other subjects, we need to understand our learning styles. The styles are namely, visual, auditory and kinesthetic.

Each of us has one particular dominant style. Knowing which one will thus serve us good. However, the learning style itself is still not enough, we need to also know the types of intelligence we possess.

The learning styles are used to gather information and ideas, through the senses, for the brain. But how do we process the information captured afterwards depends much on the intelligences that we also have.

In maths learning, what we need is the logical / mathematical intelligence.

This lets us do all sort of computations and comparisons that maths requires.

An example is the conversion of an expression in logarithmic form to its index form (and vice versa).

log_aY = X <==> a^X = Y

Though, this conversion seems simple enough, it is found that many students still have difficulties in converting the above. Do they lack this logical intelligence?

The answer is "maybe" and "maybe not".

They may have possess this logical intelligence, but has not yet enhanced it.

One way to offset this weakness in the logical comparison part of intelligence is to introduce the visual intelligence into maths learning.

What is this visual intelligence?

This is the "appearance" aspect of information processing. Here, colour, shapes, and the likes are linked up to the information.

Why introduce the visual intelligence?

The main reason is to bridge up the right and left brain, a theory that is now well-known and practiced throughout the learning and teaching communities.

Now, let's see how we can solve the logarithmic and index conversion through the use of the visual intelligence.



The examples above showed two very interesting ways to enhance the learning. :)

The first one is to replace and "beautify" the symbols with pictures or graphics to arouse retention ability.

The second one is to make use of colours to strengthen the symbols and their placements.

By seeing pictures and colours, although they are no way close to any maths topics, the learning of maths is greatly improved especially for the logical comparison portion.

Therefore, if possible, introduce as many "visuals" into maths as you can. They will make maths learning a totally new experience. Take care of intelligences and intelligences will take care of you.

Inspired ?

:=)

Indices & Logarithm - Less Mistakes with Proper Writing

Every symbols in mathematics serve certain purpose. The size of the symbols and numbers also represent unique meaning.

Many mistakes and confusions were made due to improper writing of the above two items that are very much key to mathematics.

Two of the topics that are commonly populated with writing errors are Indices and Logarithm.

Indices: a^x = Y

Logarithm: log_nA = K

From the above two examples, it is apparent that writing is crucial to give correct meaning to mathematics.

Writing therefore is an important skill while doing these two maths topics.

Discipline is needed here to maintain consistency.

What do I mean?

Let me give an example.

log₂ xy = log₂x + log₂y

The above is a correct expression written with proper size and positioning.

If writing discipline is not maintained, it may end up as:
log₂ xy = log₂x + log2y (size and position error)

Do you notice the change in meaning when size changes?

The subscript "2" representing "base 2" is conveyed as "2 times y"!

This is the result of a simple "slip" of the hand in writing the size of the "2".

Can this mistake be allowed?

I guess NO!

Why run the risk of marks being deducted or effort wasted in getting wrong answers?

Write properly is the only answer to less mistakes, if not at all.

If a^x = Y is written as ax = Y, how do we expect to get correct answers?

Here, you realise that maths learning is not just about mathematical concepts, human discipline and strength of concentration matters also.

Being able to write clearly and presentable is a noble capability that we should be proud to demonstrate to anyone. Strive to be focused and discipline when dealing with maths.

Happy writing! :P

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Merit of Grouping in Maths Solving

For those who do maths at above elementary level, you will encounter many terms involved in the already many steps to solve a mathematical question.

Example is the solution of Partial fraction, that is highly needed in calculus.

One of the steps needed is the comparing of coefficients to extract out equations to determine the numerators in the individual terms of the partial fractions.

An example is shown below.

x - 3 = A x² + B x (x + 1) + C (x - 2)

How do we solve for the unknown A, B and C?

One useful technique is to do "grouping" of relevant terms.

This is a simple yet powerful method that make the process of solving maths less confusing as it serves to gather common or liked term in the same boundaries.

What I mean is ....

From the above example, we can rewrite them as,

x - 3 = A x² + Bx (x + 1) + C (x - 2)
==> x - 3 = A x² + Bx² + Bx + Cx - 2C
==> x - 3 = (A + B)x² +(B + C)x - 2C

Here, you can see that the coefficient of the terms can be equated nicely to be :-

x² term: (A + B) = 0

x term: (B + C) = 1

Constant: 2C = 3

From herer A, B and C can be easily determined comfortably.

Thus, grouping has the ability to simplify the thinking steps due to clarity as reflected above.

Another example is in Indices simplification.

Take the example of 10ⁿ5^n/2 / 20^n/4.

We cannot see any way to combine the different base number (10, 5 and 20) unless we split them into their lowest factors. (Don't lost focus now, we are aiming for Grouping technique!)

10ⁿ = (5 x 2 )ⁿ
20ⁿ = (2 x 2 x 5)ⁿ

Rewriting to prepare for grouping,

(5 x 2 )ⁿ5^n/2 / (2 x 2 x 5)^n/4


==> 5^{(n + n/2 - n/4)} 2^{n- n/4 - n/4}

==> 5^5/4n 2^n/2

Here again, you can see the merit of grouping the common base number in order to perform the Indices operation.

Maths is "tricky" at times, but isn't this to train our mind to stay active and flexible to counter any challenges put forward. It actually enhances our self-esteem and confidence to handle problems in real life.

Happy grouping .....

:-)

Indices | Big and Small concept

In the study of indices, symbols are written with two sizes and in two different positions. They have their own unique meanings.

a² means a times a, or simply a x a.

(a + b)² means (a + b) x (a + b).

(anything)³ means (anything) x (anything) x (anything).

Therefore, from above examples, we can see that "anything" operated by a small number higher up above it means repetition by that number of times (defined by that little number).

Mistakes normally made:

(a + b)² ==> a² + b² This is wrong!
a² - b² ==> (a - b) ² This is also incorrect!

Reason

The "2" is a power and not a factor. It means repetition. And therefore cannot be factorised.

The "2" is written above the normal line (called the base), and thus has "bigger" power than the base element.

The correct answer to the mistakes:
(a + b)² = ( a + b) ( a + b) = a² + 2ab + b²
a² - b² = ( a x a ) - ( b x b ) = (a - b ) ( a + b ).

To summarise, the "smaller" number (or letter) makes the "larger" base repeats the number of times indicated by that small number.

Principles of mathematics and its indices' concept .....
Understand the principles and concepts, and you will be fine.

For more common mistakes in indices, click here.

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How To Solve Exponential Equations

Exponential equations? What are they?
They are simply equations written in the indices form xⁿ = Y.

By looking at their simple form, we can deduce that their solution will not deviate much from simplicity also.

Lets look at a few examples.

Ex 1: Solve 2^x = 16

We change the 16 to 2⁴. This allows us to compare using the mathematical logic that when the base(2) is the same, the power of the base must be the same. 2^x = 2⁴.
Therefore x = 4 (Answer).

NOTE: How about doing something like 2^x = 15? This case we need Logarithm !

Ex 2: Solving the exponential equation 2^2k - 3(2^k) + 2 = 0.

This is slightly challenging in that the exponential equation is of the quadratic form.


Maths Tips: Let the 2^k be u. And 2^2k be u². This simplifies the outlook of the exponential equation without changing its meaning.

The new modified question becomes u² - 3u +2 = 0. Is this simpler to solve? Must be!


Moving on....

Using factorisation, we get (u - 1)(u -2) = 0.
Which means (u - 1) = 0 or (u - 2) = 0.
Therefore u = 1 or u = 2. Replacing the u = 2^k back to u,
2^k = 1= 2⁰ or 2^k = 2
Answer: k = 0 or k = 1.

From the above examples, they showed that with proper planning and understanding, solving exponential equations can be simple and fun.

Therefore strive to understand the principles of the topics, and all will be fine. :)

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Indices | Commonly Made Mistakes

Humans are imperfect. Therefore to make errors in maths computation, we are reminded that we are human!

Does that mean we can excuse ourselves for making maths errors?
No...... There are maths basics that we cannot afford to make errors.

Therefore to avoid making errors in Indices, read the explanation below.
Who knows, by reading this post, you may end up one day as a professor!

Therefore to cut short everything, here it goes ......

*******************************************************************************
The format for indices are one of the simplest in maths. In its simplest form a^x = Y.

But mistakes are still being made! Why?

Firstly, the writing skill of students has to improve.
When the power of a number (base) has to be written higher up and smaller than the base number, it has to be so.
Error: a^x = Y written as ax = Y ==> the power "x" has changed meaning.

Secondly, understanding of the Laws of Indices is not fully digested.
Error: a⁰ = 1 misunderstood as a⁰ = 0 ==> concept of power is wrong as it became multiplication instead of division (e.g. a² / a² = 1).

Thirdly, the function of brackets is not fully used.
Brackets can isolate the powers when many manipulations or operations are done to simplify index terms.
Error-1: x⁴ / x ^2-n = x^4-(2-n) = x²⁺ⁿ became = x^4-2-n = x^2-n . The sign for the small "n" is wrong.

Error-2: (xK)² = x² K² taken as x K² forgetting that "x" is affected by the power 2 also.

How to solve this common indices mistakes?
Practice and practice, and clarify thoughts. This is different from learning other subjects where practicing of questions are not that intensive. Maths is a flexible and though-provoking subject that embodies many thinking skills. So Work hard and Enjoy Maths!

:-)

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How Are Indices and Logarithm Related

Is there any relationship between Indices and Logarithm?

To those unknown, Indices and logarithm are closely related to each other.
Their difference lies in the way they are presented or in their format.

Indices format: a^x = y

Logarithm format: log_aY = x

How do we select the appropriate format to use?
Which format to use depends on the initial maths problem or questions given, and what is the unknown to be solved.

If all else is given and the power of a number / expression has to be determined, select the "Logarithm" format. Power (x) = log_aY ==> directly obtaining the power x.

If the base "a" and the power "x" is known, then the Indices format can be used.

Example of Real - life Application:
In digital electronics engineering (where the base is 2 due to only two numbers present, 0 & 1), when the designer has to know the number of stages (x) to represent a count cycle of 8, he applies the logarithm format. log₂8 = x

For more information about the topics, click here to be directed to that post.

:-)

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Laws of Indices Explained

"What is Indices?" the novice to this maths topics asked.

"They are nothing but numbers with flying numeric powers!" said the maths teacher.

Here I move forward explaining the details of Indices.

Sit back and relax.......

Format of Indices is a^x = y.
NOTE: The power "x" should be written small.

There are three basic Laws of Indices:

1) Product Law of Indices

aⁿ x a^m = a ^{(n + m)}

Example: a³ x a² = a ⁵

2) Quotient Law of Indices

aⁿ / a^m = a^(n-m)

Example: a⁷ / a² = a^(7-2) = a⁵

3) Power Law of Indices

(aⁿ)^m = a^nm

Example: (a²)⁴ = a⁸

(a / b)ⁿ = aⁿ / bⁿ

Example: (K / Y)² = K² / Y²

The above are Laws that any maths students must know. However, these laws also produce sub-laws (special laws) from them. See below.

Special Laws of Indices:

• a⁰ = 1


• 1 / aⁿ = a^-n

The 2 special laws stated above are nothing new. They only have their variable replaced with the number 0, and 1 to extract commonly used operation.

:-)

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Measurement - Interstellar Distance

Measurements are done normally with objects deemed measurable.
Things like the width of a road, water level, sound level, or even amount of electrical current, are common things people measure.

But how about length that are way beyond human's grasp or sight?
I am talking about measurement out to space!

To measure parameters related to outer space, we still require measurement and mathematics, but in another form; not with the ruler, though.

The distance between star and star, star and planet, etc, are fantastically large.

Therefore, to express these long interstellar distance in the range of normal kilometre, the measurement becomes awkward.

Different units are, thus, invented to cover this aspect of distance measurement. Indices become useful in this manner.

1) Astronomical Unit (AU)

This unit is defined as the distance between the Earth to the Sun. The distance is about 150 million kilometres. (149,597,870 kilometres)

2) Light Year

This unit is defined as the distance taken by light to travel in one year.
Note however that light travels at the speed of 3 x 10⁸ metre per second. You can therefore, imagine how far is 1 light year! Only our imagination can go that far!

Example:

Distance of Solar System to nearest star is slightly more than 4 light years.
The width of our Milky Way in our galaxy is 100 thousand light years.

Remark: Light year is a measurement of distance and NOT the duration of time.

Long measurement, short measurement.....
Do you see the beauty of Maths here .....

:-)
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Logarithmic and Exponential Conversion (Video)

Logarithm and indices, or rather, the exponential form of logarithm, are closely linked,and served as a pair.

Maths learners of logarithm, thus, have to know and understand the translation of both forms (the log and exponential form).

The video here demonstrated to us how to convert the Logarithmic form to Exponential form.

This knowledge is a must for users of Logarithms (and Indices).