Question

Which of these equations is correct?

A. (2^2)^4 = 2^6

B. (3^5)^-2 = 1/3^-10

C. (5^-3)^6 = 1/5^18

D. (8^3)^2 = 8^9

Answer 1

Let's evaluate each equation one by one to determine which one is correct.

**A. \((2^2)^4 = 2^6\)**

Using the power of a power property \((a^m)^n = a^{mn}\):

\[
(2^2)^4 = 2^{2 \cdot 4} = 2^8
\]

So, the left side becomes \(2^8\), and the right side is \(2^6\).

This is incorrect since \(2^8 \neq 2^6\).

---

**B. \((3^5)^{-2} = 1/3^{-10}\)**

Starting with the left side:

\[
(3^5)^{-2} = 3^{5 \cdot -2} = 3^{-10}
\]

For the right side, using \(1/a^{-n} = a^n\):

\[
1/3^{-10} = 3^{10}
\]

This means \(3^{-10} \neq 3^{10}\).

This is incorrect since \(3^{-10} \neq 3^{10}\).

---

**C. \((5^{-3})^6 = 1/5^{18}\)**

Evaluating the left side:

\[
(5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18}
\]

For the right side, we use the property of exponents:

\[
1/5^{18} = 5^{-18}
\]

So, we have \(5^{-18} = 5^{-18}\).

This is correct.

---

**D. \((8^3)^2 = 8^9\)**

Using the power of a power property again:

\[
(8^3)^2 = 8^{3 \cdot 2} = 8^6
\]

So, the left side becomes \(8^6\) and the right side is \(8^9\).

This is incorrect since \(8^6 \neq 8^9\).

---

The correct equation is **C. \((5^{-3})^6 = 1/5^{18}\)**.