How Do You Simplify a Fraction with a Square Root?
Fractions that contain square roots appear often in algebra and geometry. Simplifying them usually involves rationalizing the denominator—rewriting so that no radical remains in the denominator. Here’s how to handle fractions with square roots.
The Absolute Best Book to Ace Your Math Test
Our Featured Products
When the Square Root Is in the Denominator
Step 1: Multiply by a Form of 1
To rationalize a denominator like √5, multiply the fraction by √5/√5 (which equals 1). Example: 3/√5 × √5/√5 = 3√5/5. The denominator is now a rational number. For practice, check our free worksheets.
Step 2: Simplify the Numerator
If the numerator has a radical, simplify it first. For 2√12/√3, simplify √12 = 2√3 to get 4√3/√3, then rationalize: 4√3/√3 × √3/√3 = 4×3/3 = 4.
When the Square Root Is in the Numerator
If only the numerator has a radical, simplify the radical and reduce common factors. Example: √18/6 = 3√2/6 = √2/2. No rationalization needed unless your teacher requires the radical in a specific place.
When Both Numerator and Denominator Have Radicals
Rationalize by multiplying by the conjugate or by the denominator’s radical. For (√2 + 1)/√3, multiply by √3/√3 to get (√6 + √3)/3. Visit Effortless Math for more algebra resources.
Frequently Asked Questions
Recommended Resources
Recommended Math Resources
Recommended Math Practice Resource
Why do we rationalize the denominator?
Convention and ease of computation. A rational denominator makes addition and comparison of fractions simpler.
What if the denominator is a sum like 1 + √2?
Multiply by the conjugate: (1 − √2)/(1 − √2). The product (1+√2)(1−√2) = 1 − 2 = −1, giving a rational denominator.
Related to This Article
More math articles
- Everything You Need to Know to Pass Algebra 1
- How to Solve a Quadratic Equation by Completing the Square?
- Top 10 SHSAT Prep Books (Our 2023 Favorite Picks)
- The Best Grade 5 ELA Practice Tests for Arizona Students
- Expected Value Calculator (Free, Step-by-Step)
- Free Grade 6 English Worksheets for Virginia Students
- The Best Grade 6 ELA Practice Tests for Oregon Students
- Decoding Data: How to Identify Representative, Random, and Biased Samples
- Free Grade 3 Math Worksheet Library for Georgia (Milestones): 49 Printable Skill Pages
- A 4-Week Grade 7 ELA Study Plan: Reading, Writing, Vocabulary, and Review That Actually Fits Real Life
What people say about "How Do You Simplify a Fraction with a Square Root? - Effortless Math"?
No one replied yet.