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Difficulty: MediumProperties of Exponents in Algebraic Expressions

If the expression (a−3b2)−2a4(b−1)3\frac{(a^{-3} b^2)^{-2}}{a^4 (b^{-1})^3} is simplified to the form axbya^x b^y, where aa and bb are positive real numbers, what is the value of the product xyxy?

  1. A
    −14-14
  2. −2-2Answer
  3. C
    22
  4. D
    1010
  5. E
    1818

Answer

−2-2
The correct answer is −2-2. Simplifying the expression step-by-step: first, apply the power of a product and power of a power rules to the numerator to get a6b−4a^6 b^{-4}. Next, simplify the denominator to get a4b−3a^4 b^{-3}. Then, apply the quotient rule to simplify the fraction to a6−4b−4−(−3)=a2b−1a^{6-4} b^{-4-(-3)} = a^2 b^{-1}. This gives x=2x=2 and y=−1y=-1. The product xyxy is 2⋅(−1)=−22 \cdot (-1) = -2.

Step-by-Step Solution

1
Simplify the numerator using the power of a power rule (xp)q=xpq(x^p)^q = x^{pq} and the power of a product rule (xy)p=xpyp(xy)^p = x^p y^p.
(a−3b2)−2=a(−3)(−2)b(2)(−2)=a6b−4(a^{-3} b^2)^{-2} = a^{(-3)(-2)} b^{(2)(-2)} = a^6 b^{-4}
To raise a product to a power, raise each factor to that power by multiplying their exponents.
2
Simplify the denominator using the power of a power rule.
a4(b−1)3=a4b(−1)(3)=a4b−3a^4 (b^{-1})^3 = a^4 b^{(-1)(3)} = a^4 b^{-3}
Multiply the exponents of bb to simplify the term (b−1)3(b^{-1})^3.
3
Divide the simplified numerator by the simplified denominator using the quotient rule xpxq=xp−q\frac{x^p}{x^q} = x^{p-q}.
a6b−4a4b−3=a6−4b−4−(−3)=a2b−1\frac{a^6 b^{-4}}{a^4 b^{-3}} = a^{6-4} b^{-4 - (-3)} = a^2 b^{-1}
Subtract the exponent of the denominator from the exponent of the numerator for each base.
4
Identify the values of xx and yy from the simplified form axbya^x b^y and calculate the product xyxy.
x=2x = 2, y=−1y = -1, so xy=2⋅(−1)=−2xy = 2 \cdot (-1) = -2
Match the simplified exponents with the variables xx and yy, then multiply them to find the final value.

Key Concept

Properties of Exponents in Algebraic Expressions

Alternative Method

Alternatively, you can rewrite the negative exponents as positive exponents by moving them across the fraction bar first. The numerator (a−3b2)−2(a^{-3} b^2)^{-2} becomes 1(a−3b2)2=1a−6b4=a6b4\frac{1}{(a^{-3} b^2)^2} = \frac{1}{a^{-6} b^4} = \frac{a^6}{b^4}. The denominator a4(b−1)3a^4 (b^{-1})^3 becomes a4b3\frac{a^4}{b^3}. Dividing these gives a6b4⋅b3a4=a2b1=a2b−1\frac{a^6}{b^4} \cdot \frac{b^3}{a^4} = \frac{a^2}{b^1} = a^2 b^{-1}. This yields x=2x = 2 and y=−1y = -1, leading to the product xy=−2xy = -2.
Estimated Time:1m 30s
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Topics

Properties of Exponents in Algebraic Expressions

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