How to Solve Radicals? (+FREE Worksheet!)
How to solve radicals, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
In this article, you learn how to simplify radicals and how to do mathematics operations with radicals.
Introduction to Square Roots: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
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
Estimate a root
- 30 is between 25 and 36.
- (sqrt{25}=5) and (sqrt{36}=6).
- So the root is between 5 and 6.
Simplify a root
- 72 has a perfect-square factor 36.
- (sqrt{72}=sqrt{36cdot2}).
- Take out the square root of 36.
Try one before moving on
Introduction to Square Roots: pop-up practice
Related Topics
- How to Solve Powers of Products and Quotients
- How to Multiply Exponents
- How to Divide Exponents
- How to Solve Zero and Negative Exponents
- How to Solve Negative Exponents and Negative Bases
Step-by-step guide to solving radicals
- A square root (radical) of (x) is a number (r) whose square is: (r^2=x)
(r) is a square root of (x). - A cube root of (x) is a number (r) whose cube is: (r^3=x)
(r) is a cube root of (x). - Radical rules: (color{blue}{sqrt[n]{a^n }=a}), (color{blue}{ sqrt{x} times sqrt{y}= sqrt{text{ xy }} } )
- We can add or subtract radicals when they have the same value under radicals: (color{blue}{sqrt{x}+sqrt{x}=2sqrt{x} } ), (color{blue}{2sqrt{x}-sqrt{x}=sqrt{x} } )
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Radicals – Example 1:
Find the square root of (sqrt{169}=)
Solution:
First factor the number: (169=13^2), Then: (sqrt{169}=sqrt{13^2 })
Now use radical rule: (color{blue}{sqrt[n]{a^n }=a})
Then: (sqrt{169}=) (sqrt{13^2 }=13)
Radicals – Example 2:
Evaluate. (sqrt{9} times sqrt{25}=)
Solution:
First factor the numbers: (9=3^2) and (25=5^2)
Then: (sqrt{9}×sqrt{25}=sqrt{3^2 }×sqrt{5^2 })
Now use radical rule: (color{blue}{sqrt[n]{a^n }=a})
Then: (sqrt{9} times sqrt{25}=) (sqrt{3^2 }×sqrt{5^2 }=3×5=15)
Radicals – Example 3:
Find the square root of (sqrt{225}=)
Solution:
First factor the number: (225=15^2), Then: (sqrt{225}=sqrt{15^2})
Now use radical rule: (color{blue}{sqrt[n]{a^n }=a})
Then: (sqrt{225}=) (sqrt{15^2}=15)
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Radicals – Example 4:
Evaluate. (2sqrt{3}-sqrt{48}= )
Solution:
There are different values under radical signs. Let’s simplify (sqrt{48}). (48) can be written as (16×3). We can write (sqrt{48}) as (sqrt{ 16×3}) or (sqrt{ 16}×sqrt{3}). (16= 4^2), Then: (sqrt{16}= sqrt{4^2}=4)
(sqrt{ 48}=4sqrt{3}). Now, we can solve (2sqrt{3}-sqrt{48}=2sqrt{3}-4sqrt{3}= -2sqrt{3})
Exercises for Solving Radicals
Find the value of each square root.
- (color{blue}{sqrt{1}})
- (color{blue}{ sqrt{4} })
- (color{blue}{ sqrt{9} })
- (color{blue}{ sqrt{900} })
- (color{blue}{ sqrt{529} })
- (color{blue}{ sqrt{90} })
Download Square Roots Worksheet
- (color{blue}{1})
- (color{blue}{2})
- (color{blue}{3})
- (color{blue}{30})
- (color{blue}{23})
- (color{blue}{3sqrt{10}})
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