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 using the power of a power rule: 3 3 4 .
Rewrite the exponent as a mixed number and use the product of powers rule: 3 1 ⋅ 3 3 1 .
Express the result in radical form: 3 3 3 . The final answer 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 options provided. The options are: 3 3 3 3 3 3 9 3 3 27 3 3
Simplifying the Expression We want to simplify 8 1 3 1 . First, we can rewrite 81 as 3 4 . So, we have ( 3 4 ) 3 1 Using the power of a power rule, we get 3 3 4 We can rewrite the exponent as a mixed number: 3 1 + 3 1 Now, using the product of powers rule, we have 3 1 ⋅ 3 3 1 Since 3 3 1 = 3 3 , the expression becomes 3 3 3
Comparing with Options Now we compare our simplified expression with the given options: 3 3 3 matches our result. 3 3 3 = 3 27 = 3 ⋅ 3 3 = 9 3 , which is not equal to 3 3 3 .
9 3 3 is not equal to 3 3 3 .
27 3 3 is not equal to 3 3 3 .
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 engineering and physics, where you might need to calculate the volume of a cube given its side length, or vice versa. For instance, if you have a cube with a volume of 81 cubic units, finding the side length involves calculating the cube root of 81, which is 8 1 3 1 . This concept also applies in financial calculations, such as determining growth rates over time.
The equivalent expression for 8 1 3 1 is 3 3 3 . This was derived by rewriting 81 as 3 4 and simplifying using properties of exponents. The correct choice is option A: 3 3 3 .
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