Question

Four students are trying to determine what number must replace the question (x^2)^7=x^4*x^8 in order to make it true

Answer 1

Answer:

Joe is correct

Step-by-step explanation:

Given equation:

$(x^2)^?=x^4 \cdot x^8$

The exponent outside the bracket is a question mark and the students are trying to determine the value of the question mark.

For ease of answering, let y be the unknown number (question mark):

$\implies (x^2)^y=x^4 \cdot x^8$

First, simplify the equation by applying exponent rules to either side of the equation:

$\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}\quad\textsf{to the left side}:$

$\implies (x^2)^y=x^4 \cdot x^8$

$\implies x^{2y}=x^4 \cdot x^8$

$\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c} \quad \textsf{to the right side}:$

$\implies x^{2y}=x^4 \cdot x^8$

$\implies x^{2y}=x^{4+8}$

$\implies x^{2y}=x^{12}$

Now, apply the exponent rule:

$x^{f(x)}=x^{g(x)} \implies f(x)=g(x)$

Therefore,

$x^{2y}=x^{12}\implies 2y=12$

Finally, solve for y:

$\implies 2y=12$

$\implies 2y \div 2 = 12 \div 2$

$\implies y=6$

Therefore, Joe is correct as the unknown number is 6.

Answer 2

Let's solve ourselves instead of believing anyone

$\\ \rm\Rrightarrow (x^2)^7=x^4x^8$

• a^m+a^n=a^m+n

$\\ \rm\Rrightarrow x^{14}=x^{4+8}$

$\\ \rm\Rrightarrow x^{14}=x^{12}$

$\\ \rm\Rrightarrow x^{14}-x^{12}=0$

$\\ \rm\Rrightarrow x^{12}(x^2-1)=0$

$\\ \rm\Rrightarrow x^2-1=0$

$\\ \rm\Rrightarrow x^2=1$

$\\ \rm\Rrightarrow x=\pm 1$

• 0 is also a solution