Which is the simplified form of $\left(\frac{2 a b}{a^{-5} b^2}\right)^{-3}$ ? Assume $a \neq 0, b \neq 0$.
A. $\frac{b^3}{8 a^{18}}$
B. $\frac{b^2}{8 a^{45}}$
C. $\frac{a^6}{4 b}$
D. $\frac{2 a^6}{b^5}$
Answer (2)
The simplified form of the expression ( a − 5 b 2 2 ab ) − 3 is 8 a 18 b 3 , which corresponds to option A. We simplified the inner fraction using exponent rules and then applied the outer exponent to reach the final result. The properties of exponents were crucial in the simplification process.
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Simplify the fraction inside the parentheses using exponent rules: a − 5 b 2 2 ab = 2 a 6 b − 1 .
Apply the outer exponent of -3 to each term: ( 2 a 6 b − 1 ) − 3 = 2 − 3 ( a 6 ) − 3 ( b − 1 ) − 3 .
Simplify each term: 2 − 3 = 8 1 , ( a 6 ) − 3 = a − 18 , and ( b − 1 ) − 3 = b 3 .
Combine the simplified terms to get the final answer: 8 a 18 b 3 .
8 a 18 b 3
Explanation
Understanding the Problem We are asked to simplify the expression ( a − 5 b 2 2 ab ) − 3 , assuming that a e q 0 and b e q 0 . We will use the properties of exponents to simplify this expression.
Simplifying the Inner Fraction First, we simplify the fraction inside the parentheses. We have a − 5 b 2 2 ab = 2 ⋅ a − 5 a ⋅ b 2 b = 2 a 1 − ( − 5 ) b 1 − 2 = 2 a 1 + 5 b − 1 = 2 a 6 b − 1 .
Applying the Outer Exponent Now, we raise this simplified expression to the power of − 3 :
( 2 a 6 b − 1 ) − 3 = 2 − 3 ( a 6 ) − 3 ( b − 1 ) − 3 .
Simplifying the Expression We know that 2 − 3 = 2 3 1 = 8 1 . Also, ( a 6 ) − 3 = a 6 × ( − 3 ) = a − 18 and ( b − 1 ) − 3 = b ( − 1 ) × ( − 3 ) = b 3 . Therefore, 2 − 3 ( a 6 ) − 3 ( b − 1 ) − 3 = 8 1 a − 18 b 3 = 8 a 18 b 3 .
Final Answer Thus, the simplified form of the given expression is 8 a 18 b 3 .
Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as physics and engineering. For example, when dealing with very large or very small numbers in scientific notation, simplifying exponents helps in calculations. Imagine you are calculating the force between two charged particles using Coulomb's law, which involves inverse square relationships. Simplifying the exponents in the distance term allows for easier computation and a clearer understanding of how the force changes with distance. This skill is also essential in computer science when analyzing algorithms and data structures, where exponential growth and decay are common.
