How to Solve Radicals? (+FREE Worksheet!)
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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