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Difficulty: MediumOperations with Variable Exponents

Simplify the following algebraic expression completely:

(5x−2y4)(−2x5y−1)2 (5x^{-2}y^4)(-2x^5y^{-1})^2

Which of the following represents the correct equivalent form?

  1. 20x8y220x^8y^2Answer
  2. B
    20x5y520x^5y^5
  3. C
    100x6y6100x^6y^6
  4. D
    −20x8y2-20x^8y^2

Answer

The correctly simplified expression is 20x8y220x^8y^2.
The correct result is achieved by following the order of operations and applying the laws of exponents. First, the exponent of 2 must be distributed to all parts of the second term: (−2)2(x5)2(y−1)2(-2)^2(x^5)^2(y^{-1})^2, which simplifies to 4x10y−24x^{10}y^{-2}. Next, this result is multiplied by the first term: (5x−2y4)(4x10y−2)(5x^{-2}y^4)(4x^{10}y^{-2}). Multiplying the coefficients gives 5×4=205 \times 4 = 20. Using the product of powers rule, adding the exponents for xx yields −2+10=8-2 + 10 = 8, and for yy yields 4+(−2)=24 + (-2) = 2. This results in the final expression 20x8y220x^8y^2.

Step-by-Step Solution

1
Apply the power of a product rule to the second term by squaring the coefficient and each variable.
(−2)2(x5)2(y−1)2=4x10y−2(-2)^2(x^5)^2(y^{-1})^2 = 4x^{10}y^{-2}
The order of operations (PEMDAS) requires evaluating exponents before performing multiplication.
2
Rewrite the full expression with the expanded term.
(5x−2y4)(4x10y−2)(5x^{-2}y^4)(4x^{10}y^{-2})
To clearly identify the coefficients and bases that need to be combined.
3
Multiply the numerical coefficients and apply the product of powers rule to the variables by adding their exponents.
(5×4)x−2+10y4+(−2)=20x8y2(5 \times 4)x^{-2+10}y^{4+(-2)} = 20x^8y^2
When multiplying exponential terms with the same base, you add the exponents.

Key Concept

Operations with Variable Exponents
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Topics

Operations with Variable Exponents

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