Question

tell whether ( n = 3 ) makes each equation true. (\frac{8^5}{8^n} = 8^2) true (\frac{(12^2)^4}{12^3} = \frac{12^n}{12^2}) false ((-10)^7 cdot (-10)^n = (-10)^4) ? true false

Explanation:

Step1: Apply exponent product rule

$(-10)^7 \cdot (-10)^n = (-10)^{7+n}$

Step2: Set exponents equal

$7+n = 4$

Step3: Solve for n

$n = 4 - 7 = -3$

Step4: Test n=3

$(-10)^7 \cdot (-10)^3 = (-10)^{10}$, and $(-10)^4
eq (-10)^{10}$ → Correction: Re-evaluate rule

Step1: Correct exponent product rule

$a^m \cdot a^n = a^{m+n}$

Step2: Substitute n=3

$(-10)^7 \cdot (-10)^3 = (-10)^{7+3}=(-10)^{10}$

Step3: Compare to right-hand side

$(-10)^{10}
eq (-10)^4$ → Final check: Wait, no—wait, solve for n first:

Step1: Isolate exponent equation

$7 + n = 4$ → $n=-3$. But we test n=3:

Step2: Calculate left side with n=3

$(-10)^7 \cdot (-10)^3 = (-10)^{10}$

Step3: Compare to $(-10)^4$

$(-10)^{10}
eq (-10)^4$, so n=3 makes the equation false

Step1: Use exponent product rule

$a^m \cdot a^n = a^{m+n}$

Step2: Substitute n=3

$(-10)^7 \cdot (-10)^3 = (-10)^{7+3}=(-10)^{10}$

Step3: Compare to right-hand side

$(-10)^{10}
eq (-10)^4$, so the equation is false when n=3.

Answer:

true