Math Problem Statement

Which option is equivalent to (9)^(-4) / (-3)^(-8)?

Solution

The expression given in the image is:

(9)4(3)8\frac{(9)^{-4}}{(-3)^{-8}}

We will simplify this expression step by step:

Step 1: Rewrite the terms with negative exponents

Recall that an=1ana^{-n} = \frac{1}{a^n}. So, the terms can be rewritten as: (9)4=194,(3)8=1(3)8(9)^{-4} = \frac{1}{9^4}, \quad (-3)^{-8} = \frac{1}{(-3)^8}

The entire expression now becomes: 1941(3)8=194×(3)8\frac{\frac{1}{9^4}}{\frac{1}{(-3)^8}} = \frac{1}{9^4} \times (-3)^8

Step 2: Simplify powers

Now calculate the powers:

  • 94=(32)4=389^4 = (3^2)^4 = 3^8, so 94=389^4 = 3^8
  • (3)8=38(-3)^8 = 3^8, because raising a negative number to an even power results in a positive number.

The expression now simplifies to: 138×38=1\frac{1}{3^8} \times 3^8 = 1

Final Answer:

The expression simplifies to 11, so the correct answer is:

A\boxed{A}

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

  1. What is the rule for handling negative exponents?
  2. How do we simplify expressions with exponents and fractions?
  3. What happens when we raise a negative base to an even power?
  4. How do we combine powers with the same base in multiplication and division?
  5. What is the difference between positive and negative exponents?

Tip: When simplifying fractions with exponents, always look for common bases to make calculations easier!

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Math Problem Analysis

Mathematical Concepts

Exponent Rules
Negative Exponents
Multiplication and Division of Powers

Formulas

a^(-n) = 1 / a^n
(-a)^n = a^n when n is even

Theorems

Properties of Exponents
Simplifying Rational Expressions

Suitable Grade Level

Grades 8-10