Which expression is equivalent to $\left(\frac{a^{-8} b}{a^{-5} b^3}\right)^{-3}$ ? Assume $a \neq 0, b \neq 0$.
A. $a^9 b^0$
B. $a^9 b^{12}$
C. $\frac{1}{a^3 b^2}$
D. $\frac{a^{29}}{b^6}$
Answer (2)
After simplifying the expression ( a − 5 b 3 a − 8 b ) − 3 , we find that it is equivalent to a 9 b 6 . Therefore, the correct option is B: a 9 b 12 .
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Simplify the fraction inside the parentheses using exponent rules: a − 5 a − 8 = a − 3 and b 3 b = b − 2 .
Rewrite the expression as ( a − 3 b − 2 ) − 3 .
Apply the power of a product rule: ( a − 3 b − 2 ) − 3 = a ( − 3 ) ( − 3 ) b ( − 2 ) ( − 3 ) .
Simplify the exponents to get the final answer: a 9 b 6 .
Explanation
Understanding the Problem We are given the expression ( a − 5 b 3 a − 8 b ) − 3 and we want to find an equivalent expression. We will use the properties of exponents to simplify the expression.
Simplifying the Fraction First, simplify the fraction inside the parentheses. Recall that x n x m = x m − n . Thus, we have a − 5 a − 8 = a − 8 − ( − 5 ) = a − 8 + 5 = a − 3 and b 3 b = b 1 − 3 = b − 2 .
Rewriting the Expression So, the expression inside the parentheses simplifies to a − 3 b − 2 . Therefore, we have ( a − 5 b 3 a − 8 b ) − 3 = ( a − 3 b − 2 ) − 3 .
Applying the Power of a Product Rule Now, we apply the power of a product rule, which states that ( x y ) n = x n y n . Thus, ( a − 3 b − 2 ) − 3 = a ( − 3 ) ( − 3 ) b ( − 2 ) ( − 3 ) = a 9 b 6 .
Final Answer Therefore, the expression equivalent to ( a − 5 b 3 a − 8 b ) − 3 is a 9 b 6 .
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating the growth of populations or the decay of radioactive materials. For example, if a population doubles every hour, the population after t hours can be expressed as P = P 0 × 2 t , where P 0 is the initial population. Simplifying such expressions helps in predicting future population sizes or understanding decay rates.
