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Difficulty: HardOperations with Variable Exponents

Connect the unsimplified exponential expressions in the left column to their corresponding reduced forms in the right column, assuming xx and yy are non-zero constants.

  • (2x−3y2)34x−5y−1 \frac{(2x^{-3}y^2)^3}{4x^{-5}y^{-1}} 2y7x4 \frac{2y^7}{x^4}
  • (3x4y−2)−2⋅(9x0y5) (3x^4y^{-2})^{-2} \cdot (9x^0y^5) y9x8 \frac{y^9}{x^8}
  • (15x3y−45x−2y3)−1 \left( \frac{15x^3y^{-4}}{5x^{-2}y^3} \right)^{-1} y73x5 \frac{y^7}{3x^5}
  • 6x−1y4⋅2x5y−23x8y−3 \frac{6x^{-1}y^4 \cdot 2x^5y^{-2}}{3x^8y^{-3}} 4y5x4 \frac{4y^5}{x^4}

Answer

Each unsimplified expression matches to a unique reduced form by carefully applying the power, product, and quotient rules of exponents.
Each expression is correctly simplified by systematically applying the product, quotient, and power rules of exponents, paying special attention to numerical coefficients and negative signs.

Step-by-Step Solution

1
Simplify the first expression by expanding the numerator's power and applying the quotient rule.
(2x−3y2)34x−5y−1=8x−9y64x−5y−1=2x−4y7=2y7x4 \frac{(2x^{-3}y^2)^3}{4x^{-5}y^{-1}} = \frac{8x^{-9}y^6}{4x^{-5}y^{-1}} = 2x^{-4}y^7 = \frac{2y^7}{x^4}
The power of a product rule applies exponents to coefficients, while the quotient rule requires subtracting denominator exponents from numerator exponents.
2
Simplify the second expression by resolving the negative outer exponent and multiplying the coefficients and variables.
(3x4y−2)−2⋅(9x0y5)=(3−2x−8y4)⋅(9⋅1⋅y5)=19x−8y4⋅9y5=x−8y9=y9x8 (3x^4y^{-2})^{-2} \cdot (9x^0y^5) = (3^{-2}x^{-8}y^4) \cdot (9 \cdot 1 \cdot y^5) = \frac{1}{9}x^{-8}y^4 \cdot 9y^5 = x^{-8}y^9 = \frac{y^9}{x^8}
Applying negative exponents to the coefficient creates a fraction, and recognizing that x0=1x^0 = 1 simplifies the second term before multiplication.
3
Simplify the third expression by resolving the fraction inside first, then applying the external negative exponent.
(15x3y−45x−2y3)−1=(3x5y−7)−1=3−1x−5y7=y73x5 \left( \frac{15x^3y^{-4}}{5x^{-2}y^3} \right)^{-1} = (3x^5y^{-7})^{-1} = 3^{-1}x^{-5}y^7 = \frac{y^7}{3x^5}
Simplifying the internal expression first reduces algebraic errors before applying the reciprocal operation for the -1 exponent.
4
Simplify the fourth expression by multiplying the numerator terms, then dividing by the denominator.
6x−1y4⋅2x5y−23x8y−3=12x4y23x8y−3=4x4−8y2−(−3)=4x−4y5=4y5x4 \frac{6x^{-1}y^4 \cdot 2x^5y^{-2}}{3x^8y^{-3}} = \frac{12x^4y^2}{3x^8y^{-3}} = 4x^{4-8}y^{2-(-3)} = 4x^{-4}y^5 = \frac{4y^5}{x^4}
Combining like terms in the numerator through multiplication organizes the expression for a straightforward application of the quotient rule.

Key Concept

Operations with Variable Exponents
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Topics

Operations with Variable Exponents

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