Question

Your friend, Margret, needs your help in solving a math problem that she got from her math teacher.

This was the problem:

What number can be put where the question mark is?
Justify your answer.
? =

Answer 1

We need to put x = -4 where the question mark lies.

Explanation:

Given the expression

`\left(x^6\right)^0=x^4\cdot \:\:x^?`

First, solve the left-hand side of the equation

`\left(x^6\right)^0`

`\mathrm{Apply\:exponent\:rule}:\quad \:a^0=1,\:\quad \:a\\e \:0`

Thus, the expression becomes

`\left(x^6\right)^0=1`

So we have to put the number which can make the right-hand side of the equation equal to 1.

Let's put x = 4 where the question mark lies and solve the equation to check whether the right-hand side of the equation becomes 1.

`x^4\cdot \:\:x^(-4)`

`\mathrm{Apply\:exponent\:rule}:\quad \:a^b\cdot \:a^c=a^(b+c)`

`=x^(4-4)`

`=x^0`

`\mathrm{Apply\:exponent\:rule}:\quad \:a^0=1,\:\quad \:a\\e \:0`

`=1`

Thus, both equations become equal when we put x = -4 on the question mark.

`\left(x^6\right)^0=x^4\cdot \:\:x^(-4)`

`1=1`

Therefore, we need to put x = -4 where the question mark lies.