Which of the following is equivalent to 16 superscript three-fourths?681264

General
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which of the following is equivalent to 16 superscript three-fourths?681264

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Let’s analyze the problem step by step to find which is equivalent to ( 16^{\frac{3}{4}} ).

Step 1: Understand the Expression ( 16^{\frac{3}{4}} )

The expression ( 16^{\frac{3}{4}} ) is a power of a power. Here’s how we can break it down:

  1. When a number is raised to the fraction ( \frac{m}{n} ), it means:

    • The denominator (( n )) determines the root.
    • The numerator (( m )) determines the power.

    So, ( 16^{\frac{3}{4}} ) can be rewritten as:

    16^{\frac{3}{4}} = \left( 16^{\frac{1}{4}} \right)^3

    This means we first find the fourth root of 16 and then raise it to the power of 3.

Step 2: Calculate the Fourth Root of 16

The fourth root of 16 (( 16^{\frac{1}{4}} )) means finding the number that gives 16 when multiplied by itself 4 times.

We know that ( 16 = 2^4 ). So:

16^{\frac{1}{4}} = \left( 2^4 \right)^{\frac{1}{4}} = 2

Thus:

16^{\frac{1}{4}} = 2

Step 3: Raise the Fourth Root to the Power of 3

Now, raise ( 16^{\frac{1}{4}} = 2 ) to the power of 3:

\left( 16^{\frac{1}{4}} \right)^3 = 2^3 = 8

Final Answer:

16^{\frac{3}{4}} = 8

Verifying Against Options

The equivalent value for ( 16^{\frac{3}{4}} ) is 8. It is not 681 or 264—only 8 satisfies the calculation.

If you have additional options or need further verification, feel free to elaborate! :blush: