Which expression is equivalent to $81^{\frac{1}{3}}$ ?
A. $3 \sqrt[3]{3}$
B. $3 \sqrt{3^3}$
C. $9 \sqrt[3]{3}$
D. $27 \sqrt[3]{3}$
Answer (2)
Rewrite 8 1 3 1 as ( 3 4 ) 3 1 .
Simplify the expression to 3 3 4 .
Rewrite 3 3 4 as 3 1 + 3 1 .
Rewrite 3 1 + 3 1 as 3 1 ⋅ 3 3 1 , which is 3 3 3 .
The equivalent expression is 3 3 3 .
Explanation
Understanding the Problem We are given the expression 8 1 3 1 and asked to find an equivalent expression from the given choices. The choices are: 3 3 3 3 3 3 9 3 3 27 3 3
Objective We want to simplify 8 1 3 1 and see which of the given options matches our simplified expression.
Simplifying the Expression First, we can rewrite 81 as 3 4 . So, we have 8 1 3 1 = ( 3 4 ) 3 1 Using the power of a power rule, we get ( 3 4 ) 3 1 = 3 3 4 Now, we can rewrite the exponent 3 4 as 1 + 3 1 . So, 3 3 4 = 3 1 + 3 1 Using the product of powers rule, we have 3 1 + 3 1 = 3 1 ⋅ 3 3 1 Since 3 3 1 is the same as 3 3 , we can write 3 1 ⋅ 3 3 1 = 3 3 3
Finding the Equivalent Expression Comparing our simplified expression 3 3 3 with the given choices, we see that it matches the first option.
Final Answer Therefore, the expression equivalent to 8 1 3 1 is 3 3 3 .
Examples
Understanding fractional exponents is useful in various fields, such as calculating growth rates or decay rates. For example, if a population grows by a factor of 81 over 3 years, then the annual growth factor is 8 1 3 1 = 3 3 3 ≈ 4.33 . This means the population roughly quadruples each year.
The expression equivalent to 8 1 3 1 is 3 3 3 , which corresponds to option A. This was derived by simplifying 81 as 3 4 and applying the rules of exponents. Thus, the correct choice from the given options is option A.
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