Math Problem Statement

Challenge  Is the value of the expression left parenthesis StartFraction 1 Over 5 Superscript negative 2 EndFraction right parenthesis Superscript 0 greater than​ 1, equal to​ 1, or less than​ 1? Use pencil and paper. If the value of the expression is greater than​ 1, show how you can change one sign to make the value less than 1. If the value is less than​ 1, show how you can change one sign to make the value greater than 1. If the value is equal to​ 1, show how you can make one change to make the value not equal to 1. Question content area bottom Part 1 The value of the expression is

Solution

To evaluate the given expression (15−2)0\left(\frac{1}{5^{-2}}\right)^0:

Step-by-Step Solution:

  1. Simplify the base of the exponent: 5−2=152=1255^{-2} = \frac{1}{5^2} = \frac{1}{25} So, 15−2=1125=25\frac{1}{5^{-2}} = \frac{1}{\frac{1}{25}} = 25

  2. Evaluate the entire expression: The given expression is (25)0\left(25\right)^0. Using the property of exponents where any nonzero number raised to the power of 0 is 1: (25)0=1\left(25\right)^0 = 1

Final Answer:

The value of the expression is equal to 1.

Adjusting the Expression:

Since the value of the expression is equal to 1, you can make one change to the expression to make it not equal to 1.

  1. Change the exponent to a non-zero value: If you change the exponent 00 to a positive or negative number, the value will no longer be 1. For example: (15−2)1=25\left(\frac{1}{5^{-2}}\right)^1 = 25 or (15−2)−1=125\left(\frac{1}{5^{-2}}\right)^{-1} = \frac{1}{25}

Follow-Up Questions:

  1. How does the rule a0=1a^0 = 1 apply to all non-zero numbers?
  2. What happens if the base inside the parentheses were zero?
  3. How does changing the negative exponent to a positive one affect the base?
  4. If you increase the exponent to 2, how does the final value of the expression change?
  5. What is the general rule for dealing with nested fractions and exponents?

Tip:

Always simplify the base fully before applying the exponent rule. This prevents errors when dealing with fractions and negative exponents.

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Negative Exponents
Zero Exponent Rule
Simplification of Expressions

Formulas

a^0 = 1 for any nonzero number a
a^(-n) = 1 / a^n for any nonzero number a and positive integer n

Theorems

Exponent Rules

Suitable Grade Level

Grades 8-10

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