Math Problem Statement

(92+8)2/3\left(\frac92+8\right)^{-2/3}

Solution

To simplify the expression (92+8)23\left(\frac{9}{2} + 8\right)^{-\frac{2}{3}}, let’s follow these steps:

  1. Calculate the inner expression:
    92+8=92+162=252\frac{9}{2} + 8 = \frac{9}{2} + \frac{16}{2} = \frac{25}{2}

  2. Rewrite the expression:
    Now, substitute 252\frac{25}{2} back into the expression: (252)23\left(\frac{25}{2}\right)^{-\frac{2}{3}}

  3. Apply the negative exponent:
    The negative exponent means we take the reciprocal: (252)23=1(252)23\left(\frac{25}{2}\right)^{-\frac{2}{3}} = \frac{1}{\left(\frac{25}{2}\right)^{\frac{2}{3}}}

  4. Evaluate the exponent 23\frac{2}{3}:
    To evaluate (252)23\left(\frac{25}{2}\right)^{\frac{2}{3}}, we can first find the cube root and then square the result:

    • The cube root of 252\frac{25}{2} is 2523\sqrt[3]{\frac{25}{2}}.
    • Squaring it, we get: (2523)2\left(\sqrt[3]{\frac{25}{2}}\right)^2

This gives the expression in simplified form.

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Math Problem Analysis

Mathematical Concepts

Exponents
Fractions
Reciprocal
Cube Roots
Simplification

Formulas

\left(a^m\right)^n = a^{m\cdot n}
a^{-m} = \frac{1}{a^m}
\sqrt[3]{a} = a^{1/3}

Theorems

Exponentiation rules
Reciprocal rule
Fraction exponent rules

Suitable Grade Level

Grades 8-10