Evaluating sixteen raised to three halves

Key concept: This problem tests rational exponents, especially the meaning of a fractional power written as a root and a power together. Once you convert the exponent correctly, the calculation is straightforward.

Step by step

Key idea

A rational exponent combines a root and a power. The rule is:

$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$

For $16^{3/2}$, the denominator 2 means square root, and the numerator 3 means cube the result.

Step-by-step method

First take the square root of 16:

$\sqrt{16} = 4$

Then raise that result to the third power:

$4^3 = 64$

So:

$16^{3/2} = 64$

You could also cube first and then take the square root:

$16^{3/2} = \sqrt{16^3} = \sqrt{4096} = 64$

Both methods give the same answer.

Why this works

The denominator in the exponent tells you the root index. A denominator of 2 means square root, 3 means cube root, and so on. The numerator tells you the power applied to the base or to the root result.

This is one of the most common rational-exponent patterns in algebra.

Common mistakes

A common mistake is to treat $16^{3/2}$ as $16^3/2$ or to multiply 16 by $\frac{3}{2}$. Exponents do not work that way. Another error is taking the square root of 16 and stopping at 4 without applying the exponent 3.

Final check

Because $16 = 4^2$, we can also write:

$16^{3/2} = (4^2)^{3/2} = 4^3 = 64$

That confirms the correct choice is 64.

Pitfall alert

The most common trap is reading the fraction exponent as a division problem. Students sometimes compute $16^3/2$ or $16 \cdot 3/2$, which has nothing to do with rational exponents. Another mistake is to take only the square root and forget the exponent on top. When you see $a^{m/n}$, remember that the bottom number names the root and the top number names the power. If you reverse that order or apply only one part of the exponent, the answer will be wrong even if the arithmetic is simple.

Try different conditions

If the problem changed to $81^{3/4}$, you would take the fourth root first: $\sqrt[4]{81}=3$, then cube it to get $27$. If it changed to $27^{2/3}$, you would take the cube root first: $\sqrt[3]{27}=3$, then square it to get $9$. These variants use the same rule $a^{m/n}=\left(\sqrt[n]{a}\right)^m$, but the root index and power both change with the fraction in the exponent.

Further reading

rational exponents, square root power rule, fractional exponent