Math Problem Statement

Simplify the expression: \( \left( \frac{a^2 b^{-3}}{x^{-1} y^2} \right)^3 \left( \frac{x^{-2} b^{-1}}{a^{3/2} y^{1/3}} \right) \)

Solution

Let's break down and simplify the expression in the image step-by-step. The expression is:

(a2b3x1y2)3(x2b1a3/2y1/3)\left( \frac{a^2 b^{-3}}{x^{-1} y^2} \right)^3 \left( \frac{x^{-2} b^{-1}}{a^{3/2} y^{1/3}} \right)

Step 1: Apply the exponent to the first fraction

We need to apply the cube (3^3) to all the terms in the first fraction:

(a2b3x1y2)3=(a2)3(b3)3(x1)3(y2)3\left( \frac{a^2 b^{-3}}{x^{-1} y^2} \right)^3 = \frac{(a^2)^3 (b^{-3})^3}{(x^{-1})^3 (y^2)^3}

Simplifying each power:

  • (a2)3=a6(a^2)^3 = a^{6}
  • (b3)3=b9(b^{-3})^3 = b^{-9}
  • (x1)3=x3(x^{-1})^3 = x^{-3}
  • (y2)3=y6(y^2)^3 = y^6

Thus, the first part simplifies to:

a6b9x3y6\frac{a^6 b^{-9}}{x^{-3} y^6}


Step 2: Multiply this result by the second fraction

We now have:

a6b9x3y6×x2b1a3/2y1/3\frac{a^6 b^{-9}}{x^{-3} y^6} \times \frac{x^{-2} b^{-1}}{a^{3/2} y^{1/3}}

We multiply the numerators and the denominators separately:

Numerator: a6b9x2b1=a6b10x2a^6 \cdot b^{-9} \cdot x^{-2} \cdot b^{-1} = a^6 \cdot b^{-10} \cdot x^{-2}

Denominator: x3y6a3/2y1/3=x3a3/2y6+1/3=x3a3/2y19/3x^{-3} \cdot y^6 \cdot a^{3/2} \cdot y^{1/3} = x^{-3} \cdot a^{3/2} \cdot y^{6 + 1/3} = x^{-3} \cdot a^{3/2} \cdot y^{19/3}


Step 3: Combine like bases

Now, we combine the powers of each base:

  • The aa terms: a6a3/2=a63/2=a9/2\frac{a^6}{a^{3/2}} = a^{6 - 3/2} = a^{9/2}
  • The bb terms: b10b^{-10} remains as it is since there is no bb in the denominator.
  • The xx terms: x2x3=x2+3=x1=x\frac{x^{-2}}{x^{-3}} = x^{-2 + 3} = x^1 = x
  • The yy terms: 1y19/3=y19/3\frac{1}{y^{19/3}} = y^{-19/3}

Step 4: Write the final simplified expression

a9/2xb10y19/3\frac{a^{9/2} \cdot x \cdot b^{-10}}{y^{19/3}}

This is the simplified result:

a9/2xb10y19/3a^{9/2} x b^{-10} y^{-19/3}


Would you like a detailed explanation of any part? Let me know if you have questions or need further clarifications!


Related Questions:

  1. How do you simplify fractional exponents like a6/3a^{6/3}?
  2. What are the rules for multiplying powers of the same base?
  3. How do negative exponents affect a fraction?
  4. What are common applications of exponent rules in algebraic expressions?
  5. How can you convert fractional exponents to radical form?

Tip: When working with powers and fractions, always apply exponent rules carefully to both the numerator and denominator.

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Math Problem Analysis

Mathematical Concepts

Exponent Rules
Algebraic Simplification
Fractional Exponents

Formulas

Power of a quotient: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \)
Multiplication of exponents: \( a^m \cdot a^n = a^{m+n} \)
Negative exponent rule: \( a^{-n} = \frac{1}{a^n} \)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 9-12