Law of Radicals

  • Laws of Radicals

Laws of Radicals


Root of a Number

The root of a number $b$ is another number which is multiplied n times to itself equals to $b$. The nth root of b $\left( \sqrt[n]{b} \right)$ is called a radical, where $r$ is the root, $b$ is the radicand and $n$ is the index or the order of the radical. It was first used by Christoff Rudolff in 1525.

Let $b$ $\in \mathfrak{R}$ and $n$ be any positive integer greater than 1.

$$ r = b^{1/n} = \sqrt[n]{b} $$

If $m$ and $n$ are positive integers, then

$$ r = b^{m/n} = \left( \sqrt[n]{b} \right)^m = \sqrt[n]{b^m} $$

It is also worth noting that if $r = b^{1/n}$, then

$$ r^n = b$$
  • $\sqrt[n]{0} = 0 $
  • If $b \neq 0$, then $b$ has a number of distinct nth roots. Some of the roots may not be real numbers.
  • If $b>0$, and $n$ is even, then $b$ has two real nth roots.
  • If $b>0$, and $n$ is odd, then $b$ has one real nth roots. This root is positive.
  • If $b \lt 0$, and $n$ is even, then $b$ has no real root.
  • If $b \lt 0$, and $n$ is odd, then $b$ has one real nth root. This root is negative.

Example:

  1. $\sqrt{64} = \pm 8$ since $8^2 = 64$ and $(-8)^2 = 64$. Therefore it has two real roots.
  2. $\sqrt[3]{64} = +4$ since $4^3 = 64$ and $(-4)^3 \neq 64$. Therefore it has only one real root.
  3. $\sqrt{-64} = 8i$ since $\sqrt{64(-1)}= 8\sqrt{-1} = 8i$. Therefore it has an imaginary root.
  4. $\sqrt[3]{-64} = -4$ since $(-4)^3 = -64$. Therefore it has one negative real root.
  5. $64^{-2/3} = \left( 64^{1/3} \right)^{-2} = \left(\sqrt[3]{64} \right)^{-2} = 4^{-2} = 1/16$


Let $a$ and $b$ be any non-zero Real number. If $n$ is even, the radicands $a$ and $b$ must be positive.

1st Law of Radicals

If a radical has an index equal to the exponent of the radicand, the result will always be the radicand itself.

$$ \sqrt[n]{a^n} = a $$

Example:

  1. $ \sqrt[3]{x^3} = x $
  2. $ \sqrt[5]{(xyz)^5} = xyz $
  3. $ \sqrt[3]{27z^3} = \sqrt[3]{3^3z^3} = 3z $

2nd Law of Radicals

A two radicals with the same indexes can be combined under the same radical.

$$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$$

Example:

  1. $\sqrt{12x^5} \cdot \sqrt{3x^3} = \sqrt{12(3)x^{5+3}} = \sqrt{36x^8} = 6x^4 $
  2. $\sqrt{8} = \sqrt{2^2 \cdot 2} = \sqrt{2^2} \cdot \sqrt{2} = 2 \sqrt{2}$

3rd Law of Radicals

When dividing two radicals with the same indexes, it can be expressed in a fraction form under the same radical, guaranteed that $b$ is not equal to zero.

$$ \sqrt[n]{ \frac{a}{b} } = \frac{ \sqrt[n]{a} }{ \sqrt[n]{b} } ; b \neq 0$$

Example:

  1. $ \frac{ \sqrt[3]{ 256x^{17} } }{ \sqrt[3]{4x^2} } $ $$\begin{align} \frac{ \sqrt[3]{ 256x^{17} } }{ \sqrt[3]{4x^2} } & = \sqrt[3]{ \frac{ 256x^{17} }{ 4x^2 } } \\ & = \sqrt[3]{ \frac{256}{4}x^{17-2} } \\ & = \sqrt[3]{ 64x^{15} } \\ & = \sqrt[3]{ 4^3(x^5)^3 } \\ & = 4x^5 \end{align} $$

4th Law of Radicals

Multiple radicals can be combined by multipying the indexes.

$$ \sqrt[m]{ \sqrt[n]{a} } = \sqrt[nm]{a}$$

Example:

  1. $ \sqrt[3] { \sqrt{x} } = \sqrt[6]{x}$
  2. $ \sqrt{ \sqrt[3]{ \sqrt[5] { 2^{15} x^{30} }} } $ $$\begin{align} \sqrt{ \sqrt[3]{ \sqrt[5]{ 2^{15} x^{30} } } } & = \sqrt[2 \cdot 3 \cdot 5]{ 2^{15}x^{30} } \\ & = \sqrt[30]{ 2^{15}x^{30} } \\ & = x \cdot 2^{15/30} \\ & = x \cdot 2^{1/2} \\ & = x \sqrt{ 2 } \end{align} $$