Operations on Radicals

  • Operations on Radicals

Operations on Radicals


Simplifying Radicals

Before operating on radical expressions, it is important to first simplify. The simplest form of radicals meet the following criteria:

  1. All factors in the radicand must be of an exponential power lower than the index.
  2. The index of the radical must as smallest as possible.
  3. There must be no fraction inside a radical sign.
  4. There must be no radical expressions in the denominator of a fraction by using rationalization. Rationalization is a process of eliminating radicals in the denominator.

Example: Simplify.

  1. $\sqrt{54x^5}$ $$\begin{align} \sqrt{54x^5} & = \sqrt{3^3 \cdot 2 \cdot x^5} & \text{(factor the radicands to the lowest prime)} \\ & = \sqrt{(3x^2)^2 \cdot 3 \cdot 2 \cdot x} & \text{(use Law I and Law II of Radicals)} \\ & = 3x^2 \sqrt{6x} \\ \end{align} $$
  2. $\sqrt[12]{64x^9y^3}$ $$\begin{align} \sqrt[12]{64x^9y^3} & = \sqrt[12]{2^6 x^9 y^3} & \text{(factor the radicands to the lowest prime)} & \\ & = \sqrt[12]{(2^2 x^3 y)^3} & \text{(use Law I and Law II of Radicals)} & \\ & = (2^2 x^3 y)^{3/12} & \text{(reduce the order of the index)} \\ & = (2^2 x^3 y)^{1/4}\\ & = \sqrt[4]{4 x^3 y}\\ \end{align} $$
  3. $\sqrt[3]{\frac{40a^3b^4}{c^6}}$
    4. $$\begin{align} \sqrt[3]{ \frac{40a^3b^4}{c^6} } & = \frac{ \sqrt[3]{40a^3b^4} }{ \sqrt[3]{c^6} } & \text{(use Law III of Radicals)} \\ & = \frac{ a \sqrt[3]{2^3 \cdot 5 \cdot b^4} }{ \sqrt[3]{(c^2)^3} } & \text{(factor the radicands to the lowest prime, then use Law I and II)} \\ & = \frac{ 2ab \sqrt[3]{5b} }{ c^2 } \\ \end{align} $$


Adding and Subtracting Radicals

When getting the sum or difference of two or more radicals, first simplify each radical and then add or subtract the coefficients of the similar radicals. Similar Radicals are radicals of the same index and radicands.

Example: Get the sums or differences.

  1. $ 2 \sqrt{200} - 3 \sqrt{50} - \sqrt{8} $ $$\begin{align} & 2 \sqrt{200} - 3 \sqrt{50} - \sqrt{8} \\ & = 2 \sqrt{10^2 \cdot 2} - 3 \sqrt{ 5^2 \cdot 2} - \sqrt{2^3} \\ & = (2 \cdot 10) \sqrt{2} - (3 \cdot 5) \sqrt{2} - 2 \sqrt{2} \\ & = (20 - 15 - 2) \sqrt{2} \\ & = 3 \sqrt{2} \\ \end{align} $$
  2. $ a \sqrt{4 \over ab} - \frac{1}{3} \sqrt{a \over b} + \frac{5}{6b} \sqrt{a^2b \over a}$

    3. Simplify the first term by rationalization. Make a fraction with the components equal to the radical in the denominator and then multiply to the equation.

    $$\begin{align} & a \sqrt{4 \over ab} \\ & = a \sqrt{4 \over ab} \cdot \frac{ \sqrt{ab} }{ \sqrt{ab} } \\ & = \frac{ 2a }{ ab } \sqrt{ab} \\ & = \frac{ 2 }{ b } \sqrt{ab} \\ \end{align} $$

    Simplifying the second term by rationalization. Use the same step as above.

    $$\begin{align} & \frac{1}{3} \sqrt{a \over b} \\ & = \frac{1}{3} \sqrt{a \over b} \cdot \frac{ \sqrt{b} }{ \sqrt{b} } \\ & = \frac{1}{3b} \sqrt{ab} \\ \end{align} $$

    Simplifying the third term.

    $$\begin{align} & \frac{5}{6b} \sqrt{a^2b \over a} \\ & = \frac{5}{6b} \sqrt{ab} \\ \end{align} $$

    Group the similar radicals.

    $$\begin{align} & \frac{2}{b} \sqrt{ab} - \frac{1}{3b} \sqrt{ab} + \frac{5}{6b} \sqrt{ab} \\ & = \left( \frac{ 2 }{ b } - \frac{1}{3b} + \frac{5}{6b} \right) \sqrt{ab} \\ & = \left( \frac{ 12 - 2 + 5 }{ 6b } \right) \sqrt{ab} \\ & = \left( \frac{ 15 }{ 6b } \right) \sqrt{ab} \\ & = \left( \frac{5}{2b} \right) \sqrt{ab} \\ \end{align} $$

Multipliying Radicals

Product of Radicals of the Same Order

When multiplying radicals of the same order or index, just multiply the coefficients of the radical and multiply the radicals using Law II of Radicals and then simplify.

Example:

  1. $5 \sqrt[3]{5x^3} \cdot 2\sqrt[3]{25x^2y^2}$ $$\begin{align} & 5 \sqrt[3]{5x^3} \cdot 2\sqrt[3]{25x^2y^2} \\ & = (5 \cdot 2) \sqrt[3]{5x^3 \cdot 25x^2y^2} \\ & = 10 \sqrt[3]{125x^5y^2} \\ & = 50x \sqrt[3]{x^2y^2} \\ \end{align} $$

Product of Radicals of Different Orders

When multiplying radicals of different orders, make the order of the radicals the same by getting the LCM of their indices.

$$ \sqrt[m]{a} \cdot \sqrt[n]{b} = a^{1/m} \cdot b^{1/n} = a^{ n/mn } \cdot b^{ m/mn } = \sqrt[mn]{a^n b^m} $$

Example:

  1. $ \sqrt{2} \sqrt[3]{3} \sqrt[4]{4} $ $$\begin{align} & \sqrt{2} \sqrt[3]{3} \sqrt[4]{4} & \text{LCM(2,3,4) = 12}\\ & = 2^{6/12} \cdot 3^{4/12} \cdot 4^{3/12} \\ & = \sqrt[12]{2^6 \cdot 3^4 \cdot 4^3} \\ & = \sqrt[12]{2^6 \cdot 3^4 \cdot (2^2)^3} \\ & = \sqrt[12]{2^{12} \cdot 3^4} \\ & = 2 \cdot 3^{4/12} \\ & = 2 \sqrt[3]{3} \\ \end{align} $$
  2. $ \sqrt[5]{x^3} \sqrt[3]{x^2} \sqrt{x} $ $$\begin{align} & \sqrt[5]{x^3} \sqrt[3]{x^2} \sqrt{x} & \text{LCM(2,3,5) = 30}\\ & = (x^3)^{6/30} \cdot (x^2)^{10/30} \cdot (x)^{15/30} \\ & = \sqrt[30]{ (x^3)^6 \cdot (x^2)^{10} \cdot x^{15} } \\ & = \sqrt[30]{ x^{18} \cdot x^{20} \cdot x^{15} } \\ & = \sqrt[30]{ x^{30} \cdot x^{23} } \\ & = x \sqrt[30]{ x^{23} }\\ \end{align} $$

Special Product of Radicals

  • $ \left( \sqrt{a} + \sqrt{b} \right) \left( \sqrt{a} - \sqrt{b} \right) = \left( \sqrt{a} \right) ^2 - \left( \sqrt{b} \right) ^2 = a - b $
  • $ \left( \sqrt{a} + \sqrt{b} \right) ^2 = \left( \sqrt{a} \right) ^2 + 2 \sqrt{a} \sqrt{b} + \left( \sqrt{b} \right) ^2 = a + 2 \sqrt{ab} + b $
  • $ \left( \sqrt[3]{a} + \sqrt[3]{b} \right) \left( \sqrt[3]{a^2} - \sqrt[3]{ab} + \sqrt[3]{b^2} \right) = \left( \sqrt[3]{a} \right) ^3 + \left( \sqrt[3]{b} \right) ^3 = a + b $
  • $ \left( \sqrt[3]{a} - \sqrt[3]{b} \right) \left( \sqrt[3]{a^2} + \sqrt[3]{ab} + \sqrt[3]{b^2} \right) = \left( \sqrt[3]{a} \right) ^3 + \left( \sqrt[3]{b} \right) ^3 = a - b $

Example:

  1. $ \sqrt{ 7 - \sqrt{2y} } \cdot \sqrt{ 7 + \sqrt{2y} } $ $$\begin{align} & \sqrt{ 7 - \sqrt{2y} } \cdot \sqrt{ 7 + \sqrt{2y} } & \text{(use Law II of Radicals)} \\ & = \sqrt{ (7 - \sqrt{2y})(7 + \sqrt{2y}) } \\ & = \sqrt{ 7^2 - (\sqrt{2y})^2 } \\ & = \sqrt{ 49 - 2y } \\ \end{align} $$
  2. $ (4 \sqrt{x} - 6 )(4 \sqrt{x} - 6 ) $ $$\begin{align} & (4 \sqrt{x} - 6 )(4 \sqrt{x} - 6 ) \\ & = (4 \sqrt{x})^2 + 2(4 \sqrt{x})(-6) + (-6)^2 \\ & = 16x - 48 \sqrt{x} + 36 \\ \end{align} $$
  3. $ \left( \sqrt[3]{3x} + \sqrt[3]{4y} \right) \left( \sqrt[3]{9x^2} - \sqrt[3]{12xy} + \sqrt[3]{16y^2} \right) $ $$\begin{align} & \left( \sqrt[3]{3x} + \sqrt[3]{4y} \right) \left( \sqrt[3]{9x^2} - \sqrt[3]{12xy} + \sqrt[3]{16y^2} \right) \\ & = \left( \sqrt[3]{3x} + \sqrt[3]{4y} \right) \left( \sqrt[3]{(3x)^2} - \sqrt[3]{(3x)(4y)} + \sqrt[3]{(4y)^2} \right) \\ & = 3x + 4y \\ \end{align} $$
  4. $ \left( \sqrt[3]{x} - \sqrt[3]{3} \right) \left( \sqrt[3]{x^2} + \sqrt[3]{3x} + \sqrt[3]{9} \right) $ $$\begin{align} & \left( \sqrt[3]{x} - \sqrt[3]{3} \right) \left( \sqrt[3]{x^2} + \sqrt[3]{3x} + \sqrt[3]{9} \right) \\ & =\left( \sqrt[3]{x} - \sqrt[3]{3} \right) \left( \sqrt[3]{x^2} + \sqrt[3]{x \cdot 3} + \sqrt[3]{3^3} \right) \\ & = x - 3 \\ \end{align} $$

Dividing Radicals

Dividing radicals has already been discussed above. To find the quotient of two or more radicals, it is important to rationalize the denominators. Here are more examples.

Example:

  1. $ \sqrt[3]{ \frac{ 54(x^4-2x^3) }{ 2(x-2) } } $ $$\begin{align} & \sqrt[3]{ \frac{ 54(x^4-2x^3) }{ 2(x-2) } } \\ & = \sqrt[3]{ \frac{ 54x^3(x-2) }{ 2(x-2) } } \\ & = \sqrt[3]{ 27x^3 } \\ & = 3x \\ \end{align} $$
  2. $ \frac{2x-3y}{\sqrt{2x}+\sqrt{3y}} $ $$\begin{align} & \frac{ 2x-3y }{ \sqrt{2x} + \sqrt{3y} } \\ & = \frac{ 2x-3y }{ \sqrt{2x}+\sqrt{3y} } \cdot \frac{ \sqrt{2x} - \sqrt{3y} }{ \sqrt{2x} - \sqrt{3y} } \\ & = \frac{ ( 2x-3y )( \sqrt{2x} - \sqrt{3y} ) }{ 2x-3y } \\ & = \sqrt{2x} -\sqrt{3y} \\ \end{align} $$
  3. $ \frac{ \sqrt{x} + 6 \sqrt{3} }{ \sqrt{x} - 2 \sqrt{3} } $ $$\begin{align} & \frac{ \sqrt{x} + 6 \sqrt{3} }{ \sqrt{x} - 2 \sqrt{3} } \\ & = \frac{ \sqrt{x} + 6 \sqrt{3} }{ \sqrt{x} - 2 \sqrt{3} } \cdot \frac{ \sqrt{x} + 2 \sqrt{3} }{ \sqrt{x} + 2 \sqrt{3} } \\ & = \frac{ ( \sqrt{x} + 6 \sqrt{3} )( \sqrt{x} + 2 \sqrt{3} ) }{ (\sqrt{x})^2 - (2 \sqrt{3})^2 } \\ & = \frac{ ( x + 2 \sqrt{3x} + 6 \sqrt{3x} + 12(3) }{ x - 4(3) } \\ & = \frac{ x + 8 \sqrt{3x} + 36 }{ x - 12 } \\ \end{align} $$
  4. $ \sqrt[3]{ \frac{ 27 }{ xy^3 } } $ $$\begin{align} & \sqrt[3]{ \frac{ 27 }{ xy^3 } } \\ & = \frac{3}{ y \sqrt[3]{x} } \cdot \frac{ \sqrt[3]{x^2} }{ \sqrt[3]{x^2} } \\ & = \frac{3 \sqrt[3]{x^2}}{xy} \\ \end{align} $$
  5. $ \frac{ 5x+3y }{ \sqrt[3]{25x^2} - \sqrt[3]{15xy} + \sqrt[3]{9y^2} } $ $$\begin{align} & \frac{ 5x+3y }{ \sqrt[3]{25x^2} - \sqrt[3]{15xy} + \sqrt[3]{9y^2} } \\ & = \frac{ 5x+3y }{ \sqrt[3]{(5x)^2} - \sqrt[3]{(5x)(3y)} + \sqrt[3]{(3y)^2} } \cdot \frac{ \sqrt[3]{5x} + \sqrt[3]{3y} }{ \sqrt[3]{5x} + \sqrt[3]{3y} } \\ & = \frac{ (5x+3y)(\sqrt[3]{5x} + \sqrt[3]{3y}) }{ 5x+3y } \\ & = \sqrt[3]{5x} + \sqrt[3]{3y} \\ \end{align} $$