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Difficulty: HardOperations with Variable Exponents
During the mathematical modeling of an electromagnetic field, an engineer derives the formula (4a3b−2)−22a−4b5\frac{(4a^3b^{-2})^{-2}}{2a^{-4}b^5} to represent the field intensity. Which of the following expressions is mathematically equivalent to this derived formula?
  1. A
    2a2b\frac{2}{a^2b}
  2. B
    a532b9\frac{a^5}{32b^9}
  3. 132a2b\frac{1}{32a^2b}Answer
  4. D
    132a10b\frac{1}{32a^{10}b}

Answer

The mathematically equivalent expression is 132a2b\frac{1}{32a^2b}.
To find the equivalent expression, first apply the power of a power rule to the numerator. Distribute the exponent -2 to each factor: 4^(-2) * (a^3)^(-2) * (b^-2)^(-2), which simplifies to (1/16) * a^-6 * b^4. Next, divide this result by the denominator, 2a^-4 * b^5. Divide the coefficients: (1/16) / 2 = 1/32. Divide the variables by subtracting the exponents of like bases: for a, it is -6 - (-4) = -2, and for b, it is 4 - 5 = -1. This yields (1/32) * a^-2 * b^-1. Finally, move the variables with negative exponents to the denominator to attain the fully simplified form.

Step-by-Step Solution

1
Apply the power of a power rule to the expression in the numerator.
(4)−2(a3)−2(b−2)−2=116a−6b4(4)^{-2}(a^3)^{-2}(b^{-2})^{-2} = \frac{1}{16}a^{-6}b^4
When an expression in parentheses is raised to a power, the outer exponent must be applied to the numerical coefficient and multiplied by the exponent of each variable inside.
2
Divide the coefficients of the numerator and the denominator.
1/162=116×12=132\frac{1/16}{2} = \frac{1}{16} \times \frac{1}{2} = \frac{1}{32}
The constant factors can be evaluated and divided separately from the variable terms.
3
Apply the quotient of powers rule for the variable bases.
a−6−(−4)=a−2a^{-6 - (-4)} = a^{-2} and b4−5=b−1b^{4 - 5} = b^{-1}
When dividing exponential terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.
4
Rewrite the expression using only positive exponents.
132a−2b−1=132a2b\frac{1}{32}a^{-2}b^{-1} = \frac{1}{32a^2b}
Final simplified algebraic expressions are standardly written by moving variables with negative exponents to the denominator.

Key Concept

Applying multiple properties of exponents, including the power of a product rule, power of a power rule, and quotient of powers rule with negative integers.
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Topics

Operations with Variable Exponents

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