What is 5 to the Power of Negative 2?

What is 5 to the Power of Negative 2?

5 to the power of -2 is equal to 1/25 or 0.04 as a decimal. When you raise a number to a negative exponent, it signifies taking the reciprocal of the number raised to the positive exponent. In this case, it means taking the reciprocal of 5 squared, which results in 1/25.

Certainly, here’s a table showing the values of 5 to the power of -2 and the corresponding decimal values:

5^(-2)	Decimal Value
1/25	0.04

So, 5 to the power of -2 is equal to 1/25 or 0.04 as a decimal.

Exponents represent an important math concept that appears throughout algebra, calculus, and other advanced mathematical fields. Exponents indicate repeated multiplication of a base number by itself a given number of times. Powers with negative exponents or negative bases require careful application of exponent rules and logic. In this post, we’ll evaluate 5 to the power of negative 2, written as 5^-2, walking through the steps and reasoning involved. We’ll also explore evaluating exponents with negative bases more broadly.

Reviewing Exponent Basics Let’s first review some fundamentals of exponents:

• Exponents indicate repeated multiplication of the base. Ex: 5^3 = 5 x 5 x 5
• Powers with negative exponents become positive exponents in the denominator. Ex: 5^-2 = 1/5^2
• Negative bases indicate the number is negative. Ex: -5^2 = (-5)(-5) = 25
• Fractional exponents take the root. Ex: 64^(1/2) = √64 = 8

Understanding these rules allows proper evaluation of more complex exponential expressions.

Evaluating 5^-2 Step-by-Step

Let’s walk through evaluating 5^-2 using exponent rules:

1. Identify -2 is a negative exponent
2. Convert to positive exponent in the denominator: 5^-2 = 1/5^2
3. Evaluate 5^2 = 5 x 5 = 25
4. Substitute 25 for 5^2: 5^-2 = 1/25

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

Exploring Negative Integer Bases

What happens when the base itself is negative, for example -5^2?

(-5)^2 = (-5)(-5) = 25

A negative base means the entire number keeps its negative sign when multiplied. So -5^2 gives a positive 25, unlike 5^-2 which becomes 1/25. Proper order of operations and paying attention to the signs is key.

Strategies for Evaluating Exponents

Some helpful strategies for evaluating exponents with negatives:

• Clearly identify whether the exponent or base is negative. This changes the steps.
• Apply exponent rules methodically step-by-step. Don’t take shortcuts.
• Use parentheses around negative bases and write signs out explicitly.
• Confirm your understanding of the order of operations. What takes precedence?
• Check your work by entering the expression into a calculator.

Regular practice builds familiarity with exponent evaluation rules and logic.

Conclusion

In this post, we evaluated 5^-2 step-by-step by applying rules for negative exponents to reach the value of 1/25. We also explored evaluating exponents with negative integer bases compared to negative exponents. Building fluency with exponents, especially those involving negatives, is important scaffolding for advanced math and science coursework. Patience and methodical practice evaluating a variety of exponential expressions will yield dividends in increased understanding and confidence. Keep chipping away at those exponent problems!