What Is X^(3/2) In Radical Form? | Radicals Made Clear

x^(3/2) equals √(x^3), which also writes as x√x when x ≥ 0.

You’ll see x^(3/2) in algebra, functions, and graphing work. It looks strange at first because the exponent is a fraction. Once you know what the numerator and denominator do, it turns into a plain radical in a clean, repeatable way.

This article shows the rewrite, then tightens it into a simplified radical form you can use on homework, quizzes, and calculators. You’ll also learn the domain detail that teachers love to test: when the expression is real, and when it’s not.

What x^(3/2) means in plain math

A fractional exponent is a compact way to mix two actions: a root and a power. The bottom number tells you the root. The top number tells you the power.

So in x^(3/2):

• The 2 in the denominator points to the square root.
• The 3 in the numerator points to the third power.

That idea is the same rule used for any rational exponent x^(m/n): take the n-th root of x, then raise the result to the m-th power. Many textbooks phrase it as:

• x^(m/n) = (n√x)^m
• x^(m/n) = n√(x^m)

Both forms match when you stay consistent about principal roots and the domain you’re working in. You’ll use both in a moment, since each one helps with a different simplification move.

Rewrite x^(3/2) into radical form in 3 moves

Start with the pattern x^(m/n) = n√(x^m). Here, m = 3 and n = 2.

1. Read the denominator as the root: denominator 2 → square root.
2. Read the numerator as the power: numerator 3 → cube.
3. Write it as a radical: x^(3/2) = √(x^3).

That’s already a correct radical form: √(x^3).

Most of the time, teachers want a simplified radical too. To simplify √(x^3), break x^3 into a perfect square times what’s left:

• x^3 = x^2 · x
• √(x^3) = √(x^2 · x) = √(x^2) · √x

Now comes the domain detail. For real-number work, √(x^2) equals |x|, not x. If your class assumes x ≥ 0 (common in early algebra lessons on rational exponents), then |x| = x, and you get:

√(x^3) = x√x (when x ≥ 0)

If your teacher does not assume x ≥ 0, keep the absolute value:

√(x^3) = |x|√x (real-number form)

Two equivalent-looking rewrites, and why both show up

You may also see x^(3/2) written as (√x)^3. That comes from the other identity x^(m/n) = (n√x)^m:

• x^(3/2) = (√x)^3
• (√x)^3 = (√x)(√x)(√x) = x√x (again, for x ≥ 0)

Both routes land in the same place under the usual real-number assumptions used in Algebra 1 and Algebra 2.

Real-number domain: when the expression is real

Because the denominator is 2, the rewrite includes a square root. Over the real numbers, √x is defined only when x ≥ 0.

That means:

• If you’re staying in real numbers, x^(3/2) is real only for x ≥ 0.
• If x is negative and you still write √x, you’ve moved into complex numbers.

This is also why many lessons quietly add “assume x ≥ 0” near the start. It keeps simplification clean and avoids absolute value surprises.

Why √(x^2) turns into |x|

The principal square root is nonnegative. So √(x^2) must also be nonnegative. When x is negative, x itself is negative, so √(x^2) cannot equal x in that case. The only expression that stays nonnegative for both positive and negative x is |x|.

So this chain is safe in real numbers:

• √(x^3) = √(x^2 · x) = √(x^2) · √x = |x|√x

Common simplification forms you can use

When a teacher asks for “radical form,” √(x^3) is the direct translation. When a teacher asks for “simplified radical form,” they usually want you to pull out perfect squares from inside the radical.

For x^(3/2), you’ll commonly use one of these, depending on the instructions:

• Direct radical form: √(x^3)
• Simplified (assuming x ≥ 0): x√x
• Simplified (real-number safe): |x|√x

For a clean reference on how rational exponents and radicals match, OpenStax lays out the same identities and cautions used in standard algebra courses. You can see that in OpenStax “Radicals and Rational Exponents”.

Check your rewrite with a quick numeric test

A fast way to catch sign errors is to plug in a value of x that keeps things real. Pick x = 4.

• Exponent form: 4^(3/2) = (4^(1/2))^3 = (2)^3 = 8
• Radical form: √(4^3) = √64 = 8
• Simplified form: 4√4 = 4 · 2 = 8

All three match. That’s the whole point of the rewrite: different notation, same value.

What if x = 0 or x = 1?

These are also good checks because they stay simple:

• 0^(3/2) = √(0^3) = √0 = 0
• 1^(3/2) = √(1^3) = √1 = 1

If your rewrite gives anything else, something went off in the exponent-to-radical step.

Table of common rational exponents and matching radicals

Use this table as a pattern matcher. If you can spot the denominator-root and numerator-power pair, you can convert almost any fractional exponent fast.

Exponent form	Radical form	Notes for simplification
x^(1/2)	√x	Real values need x ≥ 0
x^(3/2)	√(x^3)	Often simplifies to x√x when x ≥ 0
x^(5/2)	√(x^5)	Rewrite as √(x^4·x) = x^2√x (with x ≥ 0)
x^(1/3)	³√x	Cube roots stay real for negative x too
x^(2/3)	³√(x^2)	Often simplifies to (³√x)^2
x^(4/3)	³√(x^4)	Rewrite as ³√(x^3·x) = x·³√x
x^(3/4)	⁴√(x^3)	Fourth roots act like square roots for real domain
x^(-3/2)	1 / √(x^3)	Negative exponent flips to a reciprocal

What Is X^(3/2) In Radical Form? With simplification rules

If you want the direct conversion with no extra work, write √(x^3). If your teacher wants the simplest radical outside-and-inside form, factor x^3 into x^2·x and pull √(x^2) out.

Use this decision pattern:

• If the problem states x ≥ 0, you can write x√x.
• If the problem does not state a sign rule and you’re staying in real numbers, write |x|√x.
• If the class is using complex numbers, ask what form the instructor prefers, since square roots of negatives bring in i.

Khan Academy teaches the same “denominator is the root, numerator is the power” reading, with worked examples that match most school methods. See Khan Academy’s lesson on radicals and rational exponents.

Small mistakes that cost points

Fractional exponents create a few predictable traps. These are the ones that show up on tests.

Swapping the numerator and denominator

Students sometimes read x^(3/2) as “cube root, squared.” That would be x^(2/3), not x^(3/2). The denominator is the root index. The numerator is the power.

Forgetting the domain when you simplify

Going from √(x^2) to x without a sign condition is the classic slip. If no condition is given and you’re in real numbers, √(x^2) becomes |x|.

Turning √(x^3) into √x^3

The radical bar covers the whole x^3. Keep it together. √(x^3) means “take the square root after cubing.” Writing √x^3 can be read as (√x)^3 or as √(x^3), depending on formatting. On paper, it can look messy. Use parentheses to keep it clean.

Leaving a radical unsimplified when the problem asks for simplest form

Some teachers accept √(x^3) as “radical form” and still want you to simplify it. If the instructions say “simplify,” pull out x^2 from inside the square root and rewrite.

Practice set with answers

Try these to lock in the pattern. Each one follows the same read: denominator → root, numerator → power, then simplify if needed.

Practice problems

1. Write x^(1/2) in radical form.
2. Write x^(5/2) in radical form, then simplify for x ≥ 0.
3. Write x^(-3/2) in radical form.
4. Write x^(4/3) in radical form, then simplify.
5. Write (16)^(3/2) as a whole number.

Answer key

• 1) √x
• 2) √(x^5) = √(x^4·x) = x^2√x
• 3) 1/√(x^3)
• 4) ³√(x^4) = ³√(x^3·x) = x·³√x
• 5) 16^(3/2) = (√16)^3 = 4^3 = 64

Table of rewrite paths for x^(3/2)

This table shows the same expression written three ways, plus the condition that makes each line valid in real-number algebra.

Form	Rewrite	When it’s valid in real numbers
Exponent	x^(3/2)	Real values need x ≥ 0
Direct radical	√(x^3)	Real values need x ≥ 0
Simplified (common class rule)	x√x	Use when x ≥ 0 is stated or assumed
Simplified (sign-safe)	|x|√x	Use when no sign condition is stated

One last self-check before you submit

If your final answer is meant to be a radical, make sure it still equals the exponent form by testing one clean value like x = 4 or x = 9. If the values match, the rewrite is solid. If they don’t, the denominator-root or numerator-power step got flipped, or the simplification dropped a sign rule.