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.

:-) ***

Misunderstooded Power, Exponential

In maths, there are a few basic items that need to be addressed correctly before learning carries meaning.

One of them is the understanding of the function Power or the Exponential.

When you see the expression y = 2^k, what do you understand?

Is it 2 multiplied by k?

Or is it 2 multiplied k times?

This is a very common mistake or misconception by students.

2^k means 2 multiplied by k times. (or 2 to the power of k).

2 to the power of 4 means 2 x 2 x 2 x 2, which written in the exponential form, becomes 2⁴.

When this is clear, you will know that 2 x 2^k is not 4^k.
Which is yet another common mistake!

Then, what is this 2 x 2^k?

The answer is simply 2¹ x 2^k = 2^1+k.

This answer makes use of the Product Law of the index property.

To make sure you understand what I mean here, let's do another example.

5 x 2³ = ?

Solution:
It is not 10³ !
It is 5 x ( 2 x 2 x 2) ==> 5 x 8 = 40.

OK? Should be fine now, right?

Do not make this unnecessary mistake. 2^k is NOT 2 x k.

Bid farewell to this common mistake and move on..... Cheers!

Power of Using "Let" to Simplify Maths Solving

Anyone doing maths may encounter instances where the expressions are long and seemingly complex.

Example 1: 3(logx)² + 4 logx + 5 = 1

Example 2: 3(1/x)² + 4(1/x) + 5 = 1

When you jump straight into them, trying to solve them, you may encounter confusion, if inexperience with the working.

However, there is simple way to resolve this issue.

Since the mathematical expressions are complex to the eye, you can actually "simplify" them visually.

Make sure that doing any simplification, the meaning of the maths question should not change.

One technique to simplify the expression is to use "Let".

What do I mean?

Let's take the above examples to task.

Example 1: Let y = log x
The "complex" expression now becomes ==> 3 y² + 4y + 5 = 1

Example 2: Let y = 1/x
The equation becomes also ==> 3 y² + 4y + 5 = 1

See the usefulness of the technique here.

This technique is easy and familiar to anyone having learned simple algebra.
The above two expressions have been reduced to the familiar quadratic equations.

The only extra steps to complete the solution is the conversion back to find x.
This is so, since, solving the simplified equations give you the answer to "y", not x.

Thus, you have only to revert the "y" back to x through y = log x and y 1/x respectively.

Easy isn't it?
Maths is easy and interesting, if you want it to be.

:-)

Logical Solution in Maths

Maths can be solved in many ways, as one learning it knows.

There are the conventional techniques or methods type of solving.
You follow all the rules and laws, applying them diligently to solve the problem.
It is OK, not wrong.

However, sometimes by looking at the problem, you can twist a bit and deviate to use the "logical" method to short-cut the solving.

What do I mean?

Let's look at an example.

(This method of logical thinking is applied, especailly, in solving Partial Fraction.)

5 = A(x - 1) + B (x + 2)

Find A and B.

Conventional method ==> maybe simultaneous way.
Logical mehtod.....

Let x = 1, to eliminate the unknown A, and keeping the other unknown B.
5 = A(1 -1) + B (1 + 2)
==> 5 = 0 + 3B
==> B = 5 / 3 Get it?

Now to figure out getting A.
Let x = -2, to get rid of B.
5 = A(-2-1) + B(-2 + 2)
==> 5 = (-3) A + 0
==> A = - 5/3.

Finish. Short and sharp, and easy.

This is when you apply the logical part to maths solving.

Simple trick, right? Maths is Interesting. Don't forget this.
:-)

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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