Which is equivalent to $\sqrt[3]{8}^x$ ?
$\sqrt[x]{8}^3$
$8^{\frac{3}{x}}$
$8^{\frac{x}{3}}$
$8^{3 x}$
Answer (2)
Rewrite the cube root as a fractional exponent: 3 8 = 8 3 1 .
Substitute this into the original expression: ( 3 8 ) x = ( 8 3 1 ) x .
Apply the power of a power rule: ( 8 3 1 ) x = 8 3 x .
The equivalent expression is 8 3 x .
Explanation
Understanding the Problem We are given the expression 3 8 x and asked to find an equivalent expression from the list: x 8 3 , 8 x 3 , 8 3 x , 8 3 x
Rewriting the Cube Root The expression n a can be written as a n 1 . Therefore, 3 8 = 8 3 1 .
Substituting the Cube Root Now we can rewrite the given expression as ( 8 3 1 ) x .
Applying the Power of a Power Rule The expression ( a m ) n can be written as a mn . Therefore, ( 8 3 1 ) x = 8 3 1 ⋅ x = 8 3 x .
Finding the Equivalent Expression Comparing the simplified expression 8 3 x with the given options, we find that the equivalent expression is 8 3 x .
Final Answer Therefore, the expression equivalent to 3 8 x is 8 3 x .
Examples
Understanding exponential expressions like this is crucial in various fields, such as calculating compound interest, where the exponent represents the number of compounding periods. For instance, if you invest money and the interest is compounded quarterly, you would use exponents to determine the final amount. Similarly, in physics, exponential relationships are used to describe radioactive decay, where the exponent represents the time elapsed. These concepts help in making informed financial decisions and understanding natural phenomena.
The expression equivalent to 3 8 x is 8 3 x . This is determined by rewriting the cube root as a fractional exponent and applying the power of a power rule. The correct answer from the provided options is 8 3 x .
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