Question

(from unit 3, lesson 1.)

1. find the value of each variable that makes the equation true.

a. $(2^x)^4 = 2^{12}$
b. $3^5 \cdot 7^5 = e^5$
c. $5^9 \cdot 5^f = 5^4$
(from unit 3, lesson 1.)

Explanation:

Step1: Apply power of a power rule

For $(2^x)^4=2^{12}$, use $(a^m)^n=a^{mn}$.
$2^{4x}=2^{12}$
Set exponents equal: $4x=12$
Solve for $x$: $x=\frac{12}{4}=3$

Step2: Apply product of powers rule

For $3^5 \cdot 7^5=e^5$, use $a^n \cdot b^n=(ab)^n$.
$(3 \cdot 7)^5=e^5$
Calculate $3 \cdot 7=21$, so $e=21$

Step3: Apply product of powers rule

For $5^9 \cdot 5^f=5^4$, use $a^m \cdot a^n=a^{m+n}$.
$5^{9+f}=5^4$
Set exponents equal: $9+f=4$
Solve for $f$: $f=4-9=-5$

Answer:

a. $x=3$
b. $e=21$
c. $f=-5$