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

Answer:

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

Step-by-step explanation:

Given the expression

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

First, solve the left-hand side of the equation

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

$\mathrm{Apply\:exponent\:rule}:\quad \:a^0=1,\:\quad \:a\ne \:0$

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.

$x^4\cdot \:\:x^{-4}$

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

$=x^{4-4}$

$=x^0$

$\mathrm{Apply\:exponent\:rule}:\quad \:a^0=1,\:\quad \:a\ne \:0$

$=1$

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.