Math Problem Statement

Multiply and simplify the expression (8a^3b / b^5) * (5b / 4a^2).

Solution

The given problem is to multiply and simplify the following expression:

8a3bb55b4a2\frac{8a^3 b}{b^5} \cdot \frac{5b}{4a^2}

Step-by-Step Solution:

  1. Combine the two fractions: Multiply the numerators and denominators: 8a3bb55b4a2=(8a3b)(5b)(b5)(4a2)\frac{8a^3 b}{b^5} \cdot \frac{5b}{4a^2} = \frac{(8a^3 b)(5b)}{(b^5)(4a^2)}

    Simplify the numerator and denominator: (85)a3bb(4a2)b5=40a3b24a2b5\frac{(8 \cdot 5) a^3 b \cdot b}{(4a^2) b^5} = \frac{40a^3 b^2}{4a^2 b^5}

  2. Simplify the coefficients: Divide 4040 by 44: 40a3b24a2b5=10a3b2a2b5\frac{40a^3 b^2}{4a^2 b^5} = \frac{10a^3 b^2}{a^2 b^5}

  3. Simplify the variables:

    • For a3/a2a^3 / a^2, subtract the exponents: a32=a1=aa^{3-2} = a^1 = a.
    • For b2/b5b^2 / b^5, subtract the exponents: b25=b3=1b3b^{2-5} = b^{-3} = \frac{1}{b^3}.

    The simplified expression becomes: 10ab3\frac{10a}{b^3}

Final Answer:

10ab3\boxed{\frac{10a}{b^3}}

Let me know if you'd like further clarification!


Here are 5 related questions to expand your understanding:

  1. How do you handle negative exponents in simplification?
  2. What happens if one term in the numerator or denominator equals zero?
  3. How do you multiply fractions with multiple variables?
  4. What are some tips for identifying like terms when simplifying expressions?
  5. How do you simplify expressions that involve roots instead of exponents?

Tip: When simplifying fractions, always handle coefficients and variables separately to avoid confusion.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponents
Fractions

Formulas

Exponent rules: a^m / a^n = a^(m-n)
Multiplication of fractions: (a/b) * (c/d) = (a*c) / (b*d)

Theorems

Exponent subtraction rule

Suitable Grade Level

Grades 8-10