Which expression is equivalent to $\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\right)^2$?
A. $\sqrt[4]{2^3}$
B. $\sqrt{2^5}$
C. $\sqrt[4]{4^3}$
D. $\sqrt{4^5}$
Answer (2)
The expression ( 2 2 1 ⋅ 2 4 3 ) 2 simplifies to 2 5 , which corresponds to option B. The steps included combining exponents and converting to radical form. Therefore, the answer is B .
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Simplify the expression inside the parenthesis using the rule a m \t a n = a m + n , which gives 2 2 1 \t 2 4 3 = 2 4 5 .
Substitute the simplified expression back into the original expression: ( 2 4 5 ) 2 .
Simplify the expression using the rule ( a m ) n = a m \t n , which gives ( 2 4 5 ) 2 = 2 2 5 .
Rewrite the simplified expression 2 2 5 in radical form: 2 2 5 = 2 5 . The final answer is 2 5 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to ( 2 2 1 \t 2 4 3 ) 2 from the given options. The options are 4 2 3 , 2 5 , 4 4 3 , 4 5 .
Simplifying Inside Parenthesis First, we simplify the expression inside the parenthesis using the rule a m \t a n = a m + n . So, 2 2 1 \t 2 4 3 = 2 2 1 + 4 3 = 2 4 2 + 4 3 = 2 4 5
Substituting Back Next, we substitute the simplified expression back into the original expression: ( 2 4 5 ) 2
Simplifying the Power Now, we simplify the expression using the rule ( a m ) n = a m \t n . So, ( 2 4 5 ) 2 = 2 4 5 \t 2 = 2 2 5
Converting to Radical Form We rewrite the simplified expression 2 2 5 in radical form. 2 2 5 = 2 5
Finding the Matching Option Finally, we compare the simplified expression with the given options and choose the matching one. The equivalent expression is 2 5 .
Examples
Understanding exponential expressions and their simplification is crucial in various fields, such as calculating compound interest, where the amount grows exponentially over time. For instance, if you invest P dollars at an annual interest rate r compounded n times per year for t years, the final amount A can be calculated using the formula A = P ( 1 + n r ) n t . Simplifying such expressions using exponent rules helps in financial planning and investment analysis.
