Use the properties of exponents to simplify the expression: $\left(27^{\frac{1}{2}}\right)^{\frac{2}{3}}$
A) 9
B) 3
C) $\sqrt{3}$
D) 27
Answer (2)
Use the power of a power rule: ( a m ) n = a m ∗ n .
Apply the rule to the expression: ( 2 7 2 1 ) 3 2 = 2 7 2 1 × 3 2 .
Simplify the exponent: 2 1 × 3 2 = 3 1 .
Evaluate the expression: 2 7 3 1 = 3 27 = 3 . The final answer is 3 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 2 7 2 1 ) 3 2 using the properties of exponents.
Applying the Power of a Power Rule We will use the power of a power rule, which states that ( a m ) n = a m × n . Applying this rule to our expression, we get ( 2 7 2 1 ) 3 2 = 2 7 2 1 × 3 2 .
Simplifying the Exponent Now, we simplify the exponent: 2 1 × 3 2 = 2 × 3 1 × 2 = 6 2 = 3 1 .
Rewriting the Expression So our expression becomes: 2 7 3 1 .
Evaluating the Expression Finally, we evaluate the expression. Recall that a n 1 is the same as n a . Therefore, 2 7 3 1 = 3 27 = 3 , since 3 × 3 × 3 = 27 .
Final Answer Therefore, the simplified expression is 3.
Examples
Understanding exponents is crucial in many fields, such as finance and computer science. For example, calculating compound interest involves exponents. If you invest 1000 a t anann u a l in t eres t r a t eo f 5 A = P(1 + \frac{r}{n})^{nt} , w h ere P i s t h e p r in c i p a l am o u n t , r i s t h e ann u a l in t eres t r a t e , n i s t h e n u mb ero f t im es t h e in t eres t i sco m p o u n d e d p erye a r , an d t i s t h e n u mb ero f ye a rs . I n t hi sc a se , A = 1000(1 + \frac{0.05}{4})^{4 \times 5} \approx 1282.04$. Exponents are also fundamental in understanding exponential growth and decay in various scientific models.
To simplify ( 2 7 2 1 ) 3 2 , we apply the power of a power rule, which leads to 27 raised to the power of 3 1 . Evaluating this gives us the result of 3.
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