Which of these equations is correct?

Question

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

To determine which of the given equations is correct, we will evaluate each option:

A. \((2^2)^4 = 2^6\)
Using the power of a power property, \((a^m)^n = a^{mn}\), we have:
\((2^2)^4 = 2^{2 \cdot 4} = 2^8\),
which does not equal \(2^6\).
(This equation is incorrect.)

B. \((3^5)^{-2} = \frac{1}{3^{-10}}\)
Using the properties of exponents:
\((3^5)^{-2} = 3^{-10}\).
The right side, \(\frac{1}{3^{-10}} = 3^{10}\).
Since \(3^{-10} \neq 3^{10}\),
(This equation is incorrect.)

C. \((5^{-3})^6 = \frac{1}{5^{18}}\)
Using the power of a power property again:
\((5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18}\).
The right side, \(\frac{1}{5^{18}} = 5^{-18}\).
Both sides are equal; hence,
(This equation is correct.)

D. \((8^3)^2 = 8^9\)
Using the power of a power property:
\((8^3)^2 = 8^{3 \cdot 2} = 8^6\),
which does not equal \(8^9\).
(This equation is incorrect.)

Therefore, the correct equation is C. \((5^{-3})^6 = \frac{1}{5^{18}}\).