Question

Which of the following best explains how to find the value of 8^5/3?

A. A number with an exponent of 5/3 is the same as the cube root of the number raised to the 5th power
B. A number with an exponent of 5/3 is the same as 5 times the cube root of the number
C. In the table, each time x increases by 1/3, the value of 8^x is multiplied by 8
D. In the table, each time 8ˣ doubles, the exponent is multiplied by 1/3

Answer 1

The value of 8^5/3 is found by taking the cube root of 8, which is 2, and then raising it to the 5th power, resulting in 32.

Step-by-step explanation:

To find the value of 8^5/3, you need to apply the rule for fractional exponents. The fractional exponent, in this case, 5/3, can be broken down into exponentiation and a root. The original expression 8^5/3 can be understood as the cube root of 8 raised to the 5th power because the denominator of the fraction (3) represents the root, and the numerator (5) represents the power to which you would raise the result of that root. Therefore, option A is correct: A number with an exponent of 5/3 is the same as the cube root of the number raised to the 5th power.

Here’s a step-by-step approach:

1. Take the cube root of 8, which is 2, because 2 x 2 x 2 = 8.
2. Then, raise 2 to the 5th power: 2 x 2 x 2 x 2 x 2 = 32.

The value of 8^5/3 is thus 32.

Answer 2

The value of 8^5/3 is found by taking the cube root of 8 and then raising it to the 5th power, as the exponent 5/3 indicates both an exponentiation and a root.

Step-by-step explanation:

To find the value of 8^5/3, we look at the exponent 5/3 and interpret it as follows: raise 8 to the 5th power, then take the cube root of the result, or equivalently, take the cube root of 8 first and then raise the result to the 5th power. In mathematical terms, this is because an exponent that is a fraction denotes both a power and a root. Specifically, 8^5/3 is the same as the cube root of 8 raised to the 5th power (since the denominator of the fraction, 3, specifies the cube root). Therefore, the correct explanation is A. A number with an exponent of 5/3 is the same as the cube root of the number raised to the 5th power.