Math Problem Statement

solve

Rewrite \(\left(\frac{4}{3}\right)^{-3}\) without an exponent.

Solution

The problem requires rewriting the expression (43)3\left(\frac{4}{3}\right)^{-3} without an exponent. Here's the solution:

Step-by-Step Solution:

  1. Negative Exponent Rule: Recall that an=1ana^{-n} = \frac{1}{a^n}. Applying this rule, we rewrite: (43)3=1(43)3.\left(\frac{4}{3}\right)^{-3} = \frac{1}{\left(\frac{4}{3}\right)^3}.

  2. Simplify the Fraction Exponent: The cube of 43\frac{4}{3} is calculated as: (43)3=4333=6427.\left(\frac{4}{3}\right)^3 = \frac{4^3}{3^3} = \frac{64}{27}.

  3. Final Expression: Substitute this into the denominator: 1(43)3=16427.\frac{1}{\left(\frac{4}{3}\right)^3} = \frac{1}{\frac{64}{27}}.

  4. Simplify the Complex Fraction: Use the rule 1ab=ba\frac{1}{\frac{a}{b}} = \frac{b}{a}: 16427=2764.\frac{1}{\frac{64}{27}} = \frac{27}{64}.

Final Answer:

(43)3=2764.\left(\frac{4}{3}\right)^{-3} = \frac{27}{64}.

Would you like additional clarification or further details on any step?


Here are 5 related questions to expand your understanding:

  1. How would the result change if the exponent were positive ((43)3(\frac{4}{3})^3)?
  2. What is the general rule for simplifying negative exponents?
  3. How do you simplify (52)2(\frac{5}{2})^{-2}?
  4. What is the value of (43)1(\frac{4}{3})^{-1}?
  5. Can you explain how to rewrite ana^{-n} for any fraction a=pqa = \frac{p}{q}?

Tip: Always simplify exponents first, and then handle any negative signs in the exponent to avoid confusion.

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Math Problem Analysis

Mathematical Concepts

Exponents
Fractions
Negative Exponents

Formulas

\(a^{-n} = \frac{1}{a^n}\)
\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)
\(\frac{1}{\frac{a}{b}} = \frac{b}{a}\)

Theorems

Negative Exponent Rule
Fraction Exponent Simplification

Suitable Grade Level

Grades 6-8