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Difficulty: MediumOperations with Variable Exponents

Match each algebraic expression on the left with its fully simplified equivalent form on the right. Assume all variables represent non-zero real numbers.

  • 15a4b−35a−2b2\frac{15a^4b^{-3}}{5a^{-2}b^2}3a6b5\frac{3a^6}{b^5}
  • (3a−2b4)2(3a^{-2}b^4)^29b8a4\frac{9b^8}{a^4}
  • (2a3b−1)(4a−1b3)(2a^3b^{-1})(4a^{-1}b^3)8a2b28a^2b^2
  • (a2b−22)−3\left(\frac{a^2b^{-2}}{2}\right)^{-3}8b6a6\frac{8b^6}{a^6}

Answer

The quotient matches (3a^6)/b^5, the squared term matches (9b^8)/a^4, the product matches 8a^2b^2, and the fraction raised to a negative power matches (8b^6)/a^6.
Each expression is thoroughly simplified by applying the fundamental laws of exponents: adding exponents for multiplication, subtracting exponents for division, multiplying exponents for a power to a power, and reciprocating terms to handle negative exponents.

Step-by-Step Solution

1
Simplify the first expression: (15a^4b^-3) / (5a^-2b^2).
3a^6b^-5, which becomes (3a^6)/b^5.
When dividing terms with the same base, divide the coefficients and subtract the exponents. Move negative exponents to the denominator.
2
Simplify the second expression: (3a^-2b^4)^2.
9a^-4b^8, which becomes (9b^8)/a^4.
When raising a power to a power, raise the coefficient to the outer power and multiply the variable exponents.
3
Simplify the third expression: (2a^3b^-1)(4a^-1b^3).
8a^2b^2.
When multiplying terms with the same base, multiply the coefficients and add the respective exponents.
4
Simplify the fourth expression: ((a^2b^-2)/2)^-3.
a^-6b^6 / 2^-3, which becomes (8b^6)/a^6.
Apply the outer negative exponent to all factors inside the parentheses, including the denominator, then move negative powers across the fraction bar to make them positive.

Key Concept

Exponent rules: Product of powers, quotient of powers, power of a product, and negative exponents.
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Topics

Operations with Variable Exponents

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