Question

Which of these equations is correct?(1 point)
Responses

(83)2=89
open paren 8 cubed close paren squared is equal to 8 to the 9th power

(22)4=26
open paren 2 squared close paren to the 4th power is equal to 2 to the 6th power

(5−3)6=1518
open paren 5 to the negative 3 power close paren to the 6th power is equal to the fraction with numerator 1 and denominator 5 to the 18th power

(35)−2=13−10

Answer 1

Let's analyze each equation to determine whether it is correct or not.

1. **\((83)^2 = 8^9\)**:
\[
(8^3)^2 = 8^{3 \times 2} = 8^6 \quad \text{(not equal to } 8^9\text{)}
\]
Thus, this equation is **incorrect**.

2. **\((2^2)^4 = 2^6\)**:
\[
(2^2)^4 = 2^{2 \times 4} = 2^8 \quad \text{(not equal to } 2^6\text{)}
\]
Thus, this equation is **incorrect**.

3. **\((5^{-3})^6 = \frac{1}{5^{18}}\)**:
\[
(5^{-3})^6 = 5^{-3 \times 6} = 5^{-18} = \frac{1}{5^{18}} \quad \text{(this is correct)}
\]
Thus, this equation is **correct**.

4. **\((3^5)^{-2} = 1 \cdot 3^{-10}\)**:
\[
(3^5)^{-2} = 3^{5 \times -2} = 3^{-10} \quad \text{(this is equal to } 3^{-10}\text{, not } 1 \cdot 3^{-10}\text{)}
\]
Thus, this equation is **correct** (although the right side can be simplified, it is still true).

After evaluating all equations, the third equation, \((5^{-3})^6 = \frac{1}{5^{18}}\), is correct, and the fourth equation is also correct when simplified.

However, the clearest factor indicating correctness is the third equation, which does not involve any ambiguity.

So the answer is:
**(5^{-3})^6 = \frac{1}{5^{18}}** is the correct equation.