Determine whether each statement is true or false. If false, correct the right side of the equation. 5^-2 = 1/5^2

Verified step by step guidance

1. Identify the given equation: $5^{-2} = \frac{1}{5^2}$.
2. Recall the property of exponents: $a^{-n} = \frac{1}{a^n}$.
3. Apply the property to $5^{-2}$: $5^{-2} = \frac{1}{5^2}$.
4. Compare both sides of the equation: $5^{-2}$ and $\frac{1}{5^2}$.
5. Conclude that the statement is true because both sides are equal.

Key Concepts

Exponents and Negative Exponents

Exponents represent repeated multiplication of a base number. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, 5^-2 means 1/(5^2), which simplifies to 1/25. Understanding this concept is crucial for evaluating expressions involving negative exponents.

Reciprocal

The reciprocal of a number is defined as 1 divided by that number. For instance, the reciprocal of 5 is 1/5. When dealing with negative exponents, recognizing that a negative exponent signifies taking the reciprocal of the base raised to the positive exponent is essential for correctly interpreting and solving equations.

Equivalence of Expressions

Two expressions are equivalent if they yield the same value for all inputs. In this case, the statement 5^-2 = 1/5^2 is an example of equivalence, as both sides evaluate to 1/25. Understanding how to manipulate and compare expressions is vital for determining the truth of mathematical statements.