Math Problem Statement

Simplify the complex algebraic expression involving powers of numbers and variables, radicals, and fractions.

Solution

The given expression is quite complex. To solve it, we'll first break it down and simplify both the numerator and the denominator step by step.

The expression is:

(27a6b9)13(a2b581a3b4)14(64a8b10a4b2)16\frac{\left( 27a^6b^{-9} \right)^{\frac{1}{3}} \left( \frac{a^{-2}b^5}{81a^3b^{-4}} \right)^{-\frac{1}{4}}}{\left( \frac{64a^{-8}b^{10}}{a^4b^{-2}} \right)^{\frac{1}{6}}}

Step 1: Simplify powers of numbers and variables in the numerator and denominator

Numerator

  • First part: (27a6b9)13(27a^6b^{-9})^{\frac{1}{3}}

    • 2713=327^{\frac{1}{3}} = 3
    • (a6)13=a2(a^6)^{\frac{1}{3}} = a^2
    • (b9)13=b3(b^{-9})^{\frac{1}{3}} = b^{-3}

    So the first part simplifies to: 3a2b33a^2b^{-3}.

  • Second part: (a2b581a3b4)14\left( \frac{a^{-2}b^5}{81a^3b^{-4}} \right)^{-\frac{1}{4}}

    • Simplifying inside the fraction:
      • a2/a3=a5a^{-2} / a^3 = a^{-5}
      • b5/b4=b9b^5 / b^{-4} = b^9
      • 81=3481 = 3^4, so 8114=31=1381^{-\frac{1}{4}} = 3^{-1} = \frac{1}{3}

    So the second part simplifies to: (a5b981)14=13a5/4b9/4\left( \frac{a^{-5}b^9}{81} \right)^{-\frac{1}{4}} = \frac{1}{3} a^{5/4} b^{-9/4}.

Denominator

  • (64a8b10a4b2)16\left( \frac{64a^{-8}b^{10}}{a^4b^{-2}} \right)^{\frac{1}{6}}

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Math Problem Analysis

Mathematical Concepts

Exponents and Powers
Algebraic Expressions
Simplifying Fractions
Radical Expressions

Formulas

Exponent Laws: a^m * a^n = a^(m+n), (a^m)^n = a^(mn)
Radical Expressions: (a^m)^n = a^(m/n)

Theorems

Properties of Exponents
Laws of Radical Exponents

Suitable Grade Level

Grades 9-12