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Difficulty: Very hardOperations with Variable Exponents

Simplify the following algebraic expression completely. Assume all variables represent non-zero real numbers.

(−3x−2y5z−3)−2×(6x4z−1y−3)3 \left( \frac{-3 x^{-2} y^5}{z^{-3}} \right)^{-2} \times \left( \frac{6 x^4 z^{-1}}{y^{-3}} \right)^3

Which of the following represents the fully simplified expression?

  1. A
    1944x8y19z3 1944 x^8 y^{19} z^3
  2. B
    24x3y3z7 24 x^3 y^3 z^7
  3. 24x16yz9 \frac{24 x^{16}}{y z^9} Answer
  4. D
    −18x16yz9 -\frac{18 x^{16}}{y z^9}

Answer

The fully simplified expression is 24x16yz9 \frac{24 x^{16}}{y z^9} .
The correct answer is found by first distributing the outer exponents to all terms inside the parentheses (including coefficients), which yields (1/9) * x^4 * y^(-10) * z^(-6) and 216 * x^12 * z^(-3) * y^9. Multiplying these expressions by combining the coefficients and adding the exponents of like variables gives 24 * x^16 * y^(-1) * z^(-9). Rewriting with positive exponents moves y and z to the denominator, resulting in the correct simplified fraction.

Step-by-Step Solution

1
Resolve negative exponents in the denominators of both fractions to bring the variables to the numerators.
(−3x−2y5z3)−2×(6x4z−1y3)3 \left( -3 x^{-2} y^5 z^3 \right)^{-2} \times \left( 6 x^4 z^{-1} y^3 \right)^3
Moving negative exponents from the denominator to the numerator simplifies the terms before applying the outer powers.
2
Apply the power of a product rule to the first expression, multiplying each inner exponent by the outer exponent of -2.
(−3)−2x(−2)(−2)y(5)(−2)z(3)(−2)=19x4y−10z−6 (-3)^{-2} x^{(-2)(-2)} y^{(5)(-2)} z^{(3)(-2)} = \frac{1}{9} x^4 y^{-10} z^{-6}
The power of a product rule states that (abc)^x = a^x b^x c^x, and the power of a power rule states that (a^x)^y = a^(xy).
3
Apply the power of a product rule to the second expression, multiplying each inner exponent by the outer exponent of 3.
63x(4)(3)z(−1)(3)y(3)(3)=216x12z−3y9 6^3 x^{(4)(3)} z^{(-1)(3)} y^{(3)(3)} = 216 x^{12} z^{-3} y^9
Distributing the cube to the coefficient and each variable's exponent prepares the expression for multiplication.
4
Multiply the resulting expressions together by grouping coefficients and like bases.
(19×216)(x4⋅x12)(y−10⋅y9)(z−6⋅z−3) \left( \frac{1}{9} \times 216 \right) \left( x^4 \cdot x^{12} \right) \left( y^{-10} \cdot y^9 \right) \left( z^{-6} \cdot z^{-3} \right)
The commutative property of multiplication allows us to group similar terms to apply the product of powers rule.
5
Simplify the groups by multiplying coefficients and adding the exponents of like bases.
24x16y−1z−9 24 x^{16} y^{-1} z^{-9}
The product of powers rule states that a^x * a^y = a^(x+y).
6
Rewrite the expression with only positive exponents.
24x16yz9 \frac{24 x^{16}}{y z^9}
Standard simplified form requires all exponents to be positive, so terms with negative exponents are moved to the denominator.

Key Concept

Applying the power of a product, power of a power, quotient of powers, and negative exponent rules to simplify complex algebraic expressions.
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Topics

Operations with Variable Exponents

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