How to Rewrite the Expression in Radical Form

To rewrite an expression in radical form, you turn a fractional exponent like a^(m/n) into a root: a^(m/n) equals the nth root of a raised to the m power, written with a radical symbol. Radical form means an expression written with a root sign (√) instead of an exponent. The two notations are interchangeable, so you can move between them whenever one is easier to work with.

What Radical Form Means

Radical form is simply a way of writing roots using the radical symbol √. Each part of that symbol has a name. The index is the small number sitting in the notch of the sign and tells you which root you are taking — a 2 (usually left unwritten) means square root, a 3 means cube root, and so on. The radicand is the expression under the sign, the value you are taking the root of.

Radicals and exponents describe the same idea from two directions. An exponent asks “how many times do I multiply this base by itself?” A root asks the opposite question: “what number, multiplied by itself a certain number of times, gives me this value?” Because they are inverse operations, every radical can be written as a power and every fractional power can be written as a radical. That equivalence is the whole foundation for switching between the two forms, and it is covered clearly in Khan Academy’s lessons on rational exponents.

One detail matters for the rest of this article: the denominator of a fractional exponent becomes the index of the radical, and the numerator becomes the power applied to the base.

Reading the Parts of a Radical

• Index — the root you are taking (2 for square root, 3 for cube root, n for nth root).
• Radical symbol — the √ sign that signals a root.
• Radicand — everything under the sign, the quantity you take the root of.

If an expression has no visible index, it is a square root by convention. So √9 means the second root of 9, which is 3.

Convert Exponents to Radicals

The rule for rewriting a fractional exponent is:

a^(m/n) = the nth root of a^m

Read it as three moves. The denominator n becomes the index of the radical. The base a becomes the radicand. The numerator m becomes the exponent on the radicand. A quick example: x^(1/2) becomes √x, because the denominator 2 is the index and the numerator 1 means the base is simply under the root.

When the numerator is larger than 1, that power stays attached to the base inside the radical. So x^(2/3) becomes the cube root of x². You can also write it as (the cube root of x)², and both are considered correct — the exponent can sit inside or outside the radical, though inside is the more common convention. Purplemath notes that this flexibility is exactly why the two notations are treated as equivalent forms rather than competing ones.

Fractional Exponents with Coefficients

A coefficient — a number multiplying the base — stays outside the radical. For instance, 5x^(1/2) becomes 5√x. The 5 is untouched because it is not part of the base being raised to the fractional power.

If the entire product is raised to the fraction, though, the coefficient comes along. For example, (4x)^(1/2) becomes √(4x), because both the 4 and the x are under the exponent. Watch the placement of parentheses — they decide what the exponent actually applies to.

Convert Radicals to Exponents

The reverse process uses the same relationship. To turn a radical into a fractional exponent, the index becomes the denominator and any power on the radicand becomes the numerator.

Take the cube root of y⁴. The index is 3, so the denominator is 3, and the radicand carries a power of 4, so the numerator is 4. The result is y^(4/3). For a plain square root like √z, the index is 2 and the implied power is 1, giving z^(1/2).

Moving to exponents is often useful because exponent rules — multiplying powers by adding exponents, dividing by subtracting, and raising a power to a power by multiplying — are easier to apply than radical rules. Many algebra problems become simpler when you switch to exponential notation, simplify, then convert back if a radical answer is expected.

A useful check: the denominator of the exponent should always match the index of the radical you started with.

Worked Example

Example — Numbers. Write 8^(2/3) in radical form. The denominator 3 is the index, so you build a cube root. The numerator 2 becomes the power on the base 8, giving the cube root of 8². Since 8² = 64, that is the cube root of 64, which equals 4.

Common Mistakes to Avoid

• Swapping numerator and denominator. Remember: denominator = index, numerator = power on the radicand.
• Losing the coefficient. A coefficient outside the base stays outside the radical; a coefficient inside the base goes under the sign.
• Forgetting the implied index. A radical with no visible index is a square root, so its denominator is 2.
• Ignoring parentheses. The exponent applies only to what it directly touches, so use grouping symbols when the whole product is affected.

A good habit is to convert back to the original form and confirm you get the same expression. If the index and denominator match and the powers line up, your conversion is correct.

Key Takeaways

• Radical form writes roots with the √ symbol; the index gives the root and the radicand is the value under the sign.
• The core rule is a^(m/n) = the nth root of a^m, where the denominator becomes the index and the numerator becomes the power on the base.
• The reverse works too: the index of a radical becomes the denominator of the exponent, and the radicand’s power becomes the numerator.
• Coefficients outside the base stay outside the radical; coefficients inside the base move under the sign.
• Always check that your index and denominator match, and convert back once to verify the answer.