How to Simplify Radical Expressions Involving Fractions?
To simplify radical expressions involving fractions, we have two simple methods.
Tutor-style math help
Simplify Radical Expressions Involving Fractions: what to notice and how to work it
Radicals skill
Radicals are roots. Simplifying or solving with radicals is mostly about perfect powers, domain restrictions, and checking for extraneous answers.
What to notice first
Look for perfect-square, perfect-cube, or matching index factors before reaching for a calculator.
Common student mistake
Do not split a radical across addition. (sqrt{a+b}) is not usually (sqrt a+sqrt b).
Key formulas and cues
(sqrt{ab}=sqrt asqrt b)
(x^{m/n}=sqrt[n]{x^m})
(sqrt{x}text{ requires }xge0)
(text{squaring can create extraneous answers})
A reliable path
- Find perfect powersBreak the radicand into a perfect power times a leftover factor.
- Watch the domainEven roots need nonnegative radicands in real-number problems.
- Check solutionsIf you squared both sides, test answers in the original equation.
Worked examples
Simplify a radical
Example: (sqrt{72})
- 72 = 36 times 2.
- The square root of 36 is 6.
- Leave the leftover 2 inside.
Answer: (6sqrt2)
Find a radical domain
Example: (y=sqrt{x-4})
- The radicand is x – 4.
- Require x – 4 >= 0.
- Solve the inequality.
Answer: (xge4)
Try one before moving on
Try: Simplify (sqrt{98}).
Answer: (7sqrt2).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Simplify Radical Expressions Involving Fractions: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. (sqrt{50}=)
2. (sqrt{x+3}) requires:
3. After squaring both sides, you should:
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.
- (color{blue}{frac{1}{sqrt{6}}})
- (color{blue}{frac{5}{sqrt{3}}})
- (color{blue}{frac{3}{sqrt{7}-1}})
- (color{blue}{frac{8}{sqrt{5}+3}})
- (color{blue}{frac{sqrt{6}}{6}})
- (color{blue}{frac{5sqrt{3}}{3}})
- (color{blue}{frac{sqrt{7}+1}{2}})
- (color{blue}{-2sqrt{5}+6})
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