Question

which of these equations is correect

a. (5^-3)^6 = 1/5^18
b. (3^5)^-2 = 1/3^-10
c. (8^3)^2 = 8^9
d. (2^2)^4 = 2^6

Answer 1

Let's evaluate each equation one by one:

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

Using the power of a power property, we get:
\[
(5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18}
\]
And \(1/5^{18} = 5^{-18}\), so
\[
(5^{-3})^6 = 1/5^{18} \quad \text{(Correct)}
\]

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

Using the power of a power property:
\[
(3^5)^{-2} = 3^{5 \cdot -2} = 3^{-10}
\]
And \(1/3^{-10} = 3^{10}\), so
\[
(3^5)^{-2} \neq 1/3^{-10} \quad \text{(Incorrect)}
\]

c. \((8^3)^2 = 8^9\)

Using the power of a power property:
\[
(8^3)^2 = 8^{3 \cdot 2} = 8^6 \quad \text{(Incorrect, it should be \(8^6\))}
\]

d. \((2^2)^4 = 2^6\)

Using the power of a power property:
\[
(2^2)^4 = 2^{2 \cdot 4} = 2^8 \quad \text{(Incorrect, it should be \(2^8\))}
\]

In summary:
- **a** is correct.
- **b** is incorrect.
- **c** is incorrect.
- **d** is incorrect.

Only option **a** is correct.