## The Value 5^-2 is?

*The value of 5^-2 is equal to 1/5^2, which simplifies to 1/25. When you raise a number to a negative exponent, it indicates taking the reciprocal of that number raised to the positive exponent. In this case, 5^-2 is equivalent to 1/(5^2), which is 1/25.*

## Understanding Negative Exponents

In mathematics, exponents indicate repeated multiplication of a base number. For example:

53 = 5 x 5 x 5 = 125

The exponent tells us how many times to multiply the base by itself.

Negative exponents work a bit differently. A negative exponent indicates to take the reciprocal of the base, then raise it to the positive value of the exponent.

For example:

5^-2 Can be re-written as:

(1/5)^2

## Evaluating Negative Exponents

To evaluate an expression with a negative exponent:

- Rewrite the base number as a reciprocal by flipping it upside down.
- Change the exponent to positive by removing the negative sign.
- Evaluate the expression with the reciprocal base and positive exponent.

So for 5^-2:

- The reciprocal of 5 is 1/5.
- -2 becomes 2 when made positive.
- (1/5)^2 = 1/5 x 1/5 = 1/25

Therefore, the value of 5^-2 is 1/25.

## Understanding Why It Works

Negative exponents provide a compact way to indicate reciprocals and exponents together.

5^-2 = 1/(5^2)

The negative sign signals to take the reciprocal before raising to the power. This works because division and exponents are related operations.

For example:

5/5^2 = 5/25 = 1/25

So 5^-2 gives us a shortcut to indicate dividing 1 by 5 squared. Mastering negative exponents allows you to evaluate more complex exponential expressions.

## Checking Our Work

We can verify that 1/25 is the correct value for 5^-2 by entering it into a calculator. Typing in 5 followed by the ^ symbol and -2 will give 0.04, which is 1/25 in decimal form. Getting the same result confirms our understanding of how to evaluate expressions with negative exponents.

In summary, negative exponents indicate to take the reciprocal of the base first before raising it to the corresponding positive power. This provides mathematical shorthand for division involving exponents.

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