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.
- \(\sqrt{72}=\sqrt{36\cdot2}\).
- 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: \(\color{blue}{\sqrt{x}+\sqrt{x}=2\sqrt{x} } \), \(\color{blue}{2\sqrt{x}-\sqrt{x}=\sqrt{x} } \)
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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. \(2\sqrt{3}-\sqrt{48}= \)
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: \(\sqrt{16}= \sqrt{4^2}=4\)
\(\sqrt{ 48}=4\sqrt{3}\). Now, we can solve \(2\sqrt{3}-\sqrt{48}=2\sqrt{3}-4\sqrt{3}= \ -2\sqrt{3}\)
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
- \(\color{blue}{1}\)
- \(\color{blue}{2}\)
- \(\color{blue}{3}\)
- \(\color{blue}{30}\)
- \(\color{blue}{23}\)
- \(\color{blue}{3\sqrt{10}}\)
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