How to Simplify Radical Expressions Involving Fractions?

A radical contains an expression that is not a perfect root it is called an irrational number. To rationalize the denominator, you need to get rid of all radicals that are in the denominator.

Related Topics

• How to Rationalize Radical Expressions
• How to Simplify Radical Expressions
• How to Multiply Radical Expressions

A Step-by-Step Guide to Simplifying Radical Expressions Involving Fraction

To simplify radical expressions involving fractions:

• If there is a radical in the denominator, multiply the numerator and denominator by the radical in the denominator.
• If there is a radical and another integer in the denominator, multiply both the numerator and denominator by the conjugate of the denominator.

Simplifying Radical Expressions Involving Fractions – Example 1:

Simplify. \(\frac{1}{\sqrt{7}}\)

Solution:

Multiply by the \(\sqrt{7}\): \(\frac{1}{\sqrt{7}} × \frac{\sqrt{7}}{\sqrt{7}}= \frac{\sqrt{7}} {7}\)

Simplifying Radical Expressions Involving Fractions – Example 2:

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Simplify. \(\frac{2}{\sqrt{3}+1}\)

Solution:

Multiply by the conjugate: \(\frac{\sqrt{3}-1} {\sqrt{3}-1}\)

\(\frac{2}{\sqrt{3}+1} × \frac{\sqrt{3}-1} {\sqrt{3}-1}=\frac{2(\sqrt{3}-1)}{2}=\sqrt{3}-1\)

Exercises for Simplifying Radical Expressions Involving Fractions

Simplify radical expressions.

1. \(\color{blue}{\frac{1}{\sqrt{6}}}\)
2. \(\color{blue}{\frac{5}{\sqrt{3}}}\)
3. \(\color{blue}{\frac{3}{\sqrt{7}-1}}\)
4. \(\color{blue}{\frac{8}{\sqrt{5}+3}}\)

1. \(\color{blue}{\frac{\sqrt{6}}{6}}\)
2. \(\color{blue}{\frac{5\sqrt{3}}{3}}\)
3. \(\color{blue}{\frac{\sqrt{7}+1}{2}}\)
4. \(\color{blue}{-2\sqrt{5}+6}\)

Square Root of a Fraction: Split, Simplify, Rationalize

When the whole fraction sits under the radical, work in three steps.

1. Split the root. For a ≥ 0 and b > 0, √(a/b) = √a / √b.
2. Simplify each part. Pull out any perfect square factors.
3. Rationalize. If a root is still in the denominator, multiply the top and bottom by a factor that makes the bottom a whole number.

Example 1: √(8/25) = √8 / √25 = 2√2 / 5. The denominator is already a whole number, so you are done.

Example 2: √(3/8) = √3 / √8 = √3 / (2√2). Multiply the top and bottom by √2 to get √6 / (2 × 2) = √6 / 4.

Example 3 (cube root): Simplify 6/∛2. Multiplying by ∛2 only gives ∛4 in the denominator, which is still not a whole number. You need ∛8, so multiply the top and bottom by ∛4: 6∛4 / ∛8 = 6∛4 / 2 = 3∛4.

Conjugates and Common Mistakes

If the denominator has two terms with a square root, multiply by its conjugate. The middle terms cancel, so the root disappears.

Example: 6/(3 − √3) × (3 + √3)/(3 + √3) = 6(3 + √3)/(9 − 3) = 6(3 + √3)/6 = 3 + √3.

Watch out for these mistakes:

• Multiplying only the denominator. Whatever you multiply the bottom by, you must also multiply the top by.
• Splitting a sum. √(9 + 16) is 5, not 3 + 4.
• Stopping too early. Always check whether the final numerator and denominator share a common factor.

Frequently Asked Questions

How do you simplify radicals with fractions?

Split the root over the top and bottom, √(a/b) = √a/√b, then simplify each part. For √(18/25), write √18/√25 = 3√2/5. If a root remains in the denominator, multiply the top and bottom by that root, so 1/√3 becomes √3/3.

How do you simplify radical expressions with fractions and variables?

Treat the variables like numbers: pull out perfect-square factors and split the fraction. For √(x³/16) with x ≥ 0, write √(x² · x)/√16 = x√x/4. With a variable in the denominator, such as √(5/y), rationalize: √5/√y = √(5y)/y, assuming y > 0.

Can you simplify fractions with radicals?

Yes, but first simplify the radical, then divide every term in the numerator by a common factor. For (6 + √12)/2, write √12 = 2√3 to get (6 + 2√3)/2 = 3 + √3. You cannot cancel a number inside a root with one outside it, so √12/2 is not √6.

What is a fully simplified radical expression?

A radical expression is fully simplified when the radicand has no perfect-square factors (for square roots), there is no fraction under the root, and there is no radical left in the denominator. So √(8/9) is not done yet, but 2√2/3 is.

How do you simplify radical expressions by rationalizing the denominator?

Multiply the top and bottom by a factor that clears the root. For a single root, use that root: 5/√2 = 5√2/2. For a two-term denominator, use the conjugate: 3/(2 + √3) × (2 − √3)/(2 − √3) = 3(2 − √3)/(4 − 3) = 6 − 3√3.

How do you simplify radical expressions with fractions and exponents?

Rewrite the fractional exponent as a root: a^(m/n) means the nth root of a, raised to the m. For (8/27)^(2/3), take the cube root first to get 2/3, then square it to get 4/9. Taking the root before the power keeps the numbers small.