Radicals

Radicals

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

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: \(\color{blue}{\sqrt{x}+\sqrt{x}=2\sqrt{x} } \), \(\color{blue}{2\sqrt{x}-\sqrt{x}=\sqrt{x} } \)

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

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: \(\sqrt{3^2 }×\sqrt{5^2 }=3×5=15\)

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

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

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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