How to Solve Radicals

How to Solve Radicals

In this article, you learn how to simplify radicals and how to do mathematics operations with radicals.

Related Topics

Step by step guide to solve 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: \(\color{blue}{\sqrt[n]{a^n }=a}\), \(\color{blue}{ \sqrt{x} \times \sqrt{y}= \sqrt{xy} } \)
  • We can add or subtract radicals when they have exactly the same value under radicals: \(\color{blue}{\sqrt{x}+\sqrt{x}=2\sqrt{x} } \), \(\color{blue}{2\sqrt{x}-\sqrt{x}=\sqrt{x} } \)

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: \(\sqrt{13^2 }=13\)

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: \(\sqrt{3^2 }×\sqrt{5^2 }=3×5=15\)

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: \(\sqrt{15^2}=15\)

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 \(\sqrt{ 16}×\sqrt{3}\). \(\sqrt{ 16}=4\), then:
\(\sqrt{ 48}=4\sqrt{3}\). 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 each square root.

  1. \(\color{blue}{\sqrt{1}}\)
  2. \(\color{blue}{ \sqrt{4} }\)
  3. \(\color{blue}{ \sqrt{9} }\)
  4. \(\color{blue}{ \sqrt{900} }\)
  5. \(\color{blue}{ \sqrt{529} }\)
  6. \(\color{blue}{ \sqrt{90} }\)

Download Square Roots Worksheet

  1. \(\color{blue}{1}\)
  2. \(\color{blue}{2}\)
  3. \(\color{blue}{3}\)
  4. \(\color{blue}{30}\)
  5. \(\color{blue}{23}\)
  6. \(\color{blue}{3\sqrt{10}}\)

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