Math  /  Algebra

QuestionQuestion
Which expression is equivalent to (51)2?\left(5^{-1}\right)^{2} ?
Answer 125\frac{1}{25} 125 5 1125\frac{1}{125}

Studdy Solution

STEP 1

1. We are given the expression (51)2\left(5^{-1}\right)^{2}.
2. We need to simplify this expression to find an equivalent expression.
3. The rules of exponents will be used to simplify the expression.

STEP 2

1. Apply the power of a power rule for exponents.
2. Simplify the expression to find the equivalent expression.

STEP 3

Apply the power of a power rule for exponents, which states that (am)n=amn(a^m)^n = a^{m \cdot n}.
For the expression (51)2\left(5^{-1}\right)^{2}, apply the rule:
(51)2=512=52\left(5^{-1}\right)^{2} = 5^{-1 \cdot 2} = 5^{-2}

STEP 4

Simplify 525^{-2} using the definition of negative exponents, which states that an=1ana^{-n} = \frac{1}{a^n}.
52=1525^{-2} = \frac{1}{5^2}

STEP 5

Calculate 525^2:
52=255^2 = 25
Therefore,
152=125\frac{1}{5^2} = \frac{1}{25}
The equivalent expression is:
125\boxed{\frac{1}{25}}

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