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Multiplication & Division with Radicals

(Multiplication & Division with Surds)


We now learn how to multiply & divide radicals, or surds.

By the end of this section we'll know how to write expressions like the ones shown below, in their simplest form:

\[3\sqrt{2}\times \sqrt{5}\] \[\frac{2\sqrt{18}}{\sqrt{3}}\] \[5\sqrt{7}.\begin{pmatrix} 2\sqrt{3} - \sqrt{5}\end{pmatrix}\] \[\vdots \]

Tutorial 1

(Operations with Radicals)

In this first tutorial we learn about the "basic" operations with radicals.
In particular we learn about:

  • addition and subtraction
  • multiplication and division
We'll learn that: \[\sqrt{a}+\sqrt{b} \neq \sqrt{a+b}\] and: \[\sqrt{a}-\sqrt{b} \neq \sqrt{a-b}\] We also introduce the formula we work with in this section, those are the formula for multiplication and division with radicals, those are: \[\sqrt{a}\times \sqrt{b} = \sqrt{a\times b}\]
and
\[\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\]



Multiplication

Two radicals \(\sqrt{a}\) and \(\sqrt{b}\) can be multiplied together using the following formula: \[\sqrt{a} \times \sqrt{b} = \sqrt{a\times b }\] Example:
We multiply the radicals \(\sqrt{5}\) and \(\sqrt{7}\) as follows: \[\begin{aligned} \sqrt{5}\times \sqrt{7} &= \sqrt{5 \times 7} \\ & = \sqrt{35} \end{aligned}\]

Tutorial

In the following tutorial we learn how to use the formula we learnt above.
In particular we see how to write each of the expressions, below, in the simplest form:

  • \(\sqrt{2} \times \sqrt{7}\)
  • \(3\sqrt{2}\times \sqrt{5}\)
  • \(\sqrt{2}.\begin{pmatrix}\sqrt{5}+\sqrt{3}\end{pmatrix}\)
  • \(\begin{pmatrix} \sqrt{5}-\sqrt{3} \end{pmatrix}.\begin{pmatrix} \sqrt{2}+\sqrt{7}\end{pmatrix}\)



Exercise 1

Simplfy each of the following:

  1. \(\sqrt{3}\times \sqrt{7}\)
  2. \(4\sqrt{2}\times\sqrt{5}\)
  3. \(5\sqrt{3}\times 2\sqrt{11}\)
  4. \(2\sqrt{3}\times \sqrt{5} \times \sqrt{7}\)
  5. \(3\sqrt{11}\times \frac{\sqrt{5}}{2}\)

Answers Without Working

Exercise 2

Simplify each of the following:

  1. \(\sqrt{2}.\begin{pmatrix}\sqrt{3}+\sqrt{5}\end{pmatrix}\)
  2. \(3\sqrt{5}.\begin{pmatrix}2\sqrt{3} - 5\sqrt{7}\end{pmatrix}\)
  3. \(\begin{pmatrix} \sqrt{3} - \sqrt{5} \end{pmatrix}.\begin{pmatrix} \sqrt{2}+ \sqrt{7}\end{pmatrix}\)
  4. \(\begin{pmatrix}2\sqrt{11}+ \sqrt{3}\end{pmatrix}.\begin{pmatrix}\sqrt{6}-3\sqrt{2}\end{pmatrix}\)
  5. \(\begin{pmatrix}5\sqrt{10}+2\sqrt{3}\end{pmatrix}.\begin{pmatrix}3\sqrt{7} - 7\sqrt{2}\end{pmatrix}\)

Exercise

Write all the tearms in each of thefollowing expansions and simplify as much as possible:

  1. \(\begin{pmatrix}\sqrt{5}+\sqrt{7}\end{pmatrix}^2\)
  2. \(\begin{pmatrix}\sqrt{5}-\sqrt{7}\end{pmatrix}^2\)
  3. \(\begin{pmatrix}3\sqrt{10}+\sqrt{5}\end{pmatrix}^2\)
  4. \(\begin{pmatrix}\sqrt{11}-3\sqrt{6}\end{pmatrix}^2\)
  5. \(\begin{pmatrix}7\sqrt{2}+3\sqrt{5}\end{pmatrix}^2\)
  6. \(\begin{pmatrix}5\sqrt{6}-7\sqrt{10}\end{pmatrix}^2\)
  7. \(\begin{pmatrix}5\sqrt{2} + 3\sqrt{7}\end{pmatrix}^2\)
  8. \(\begin{pmatrix}8\sqrt{10} - 5\sqrt{13}\end{pmatrix}^2\)

Division

Two radicals \(\sqrt{a}\) and \(\sqrt{b}\) can be divided using the following formula: \[\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\] Example
We simplify the expression \(\frac{\sqrt{15}}{\sqrt{3}}\) as follows: \[\begin{aligned} \frac{\sqrt{15}}{\sqrt{3}} & = \sqrt{\frac{15}{3}} \\ & = \sqrt{5}\end{aligned}\]

Tutorial

In the following tutorial we learn how to use the formula we've just seen, \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).
In particular we see how to simplify each of the following:

  • \(\frac{\sqrt{10}}{4\sqrt{2}}\)
  • \(\frac{\sqrt{128}}{3\sqrt{2}}\)
  • \(\frac{2\sqrt{27}}{5\sqrt{3}}\)



Exercise 3

Simplify each of the following:

  1. \(\frac{\sqrt{35}}{\sqrt{5}}\)
  2. \(\frac{\sqrt{32}}{\sqrt{2}}\)
  3. \(\frac{2\sqrt{75}}{\sqrt{3}}\)
  4. \(\frac{\sqrt{66}}{2\sqrt{11}}\)
  5. \(\frac{\sqrt{128}}{3\sqrt{8}}\)
  6. \(\frac{4\sqrt{10}}{\sqrt{5}}\)
  7. \(\frac{6\sqrt{2}}{\sqrt{8}}\)
  8. \(\frac{\sqrt{32}}{\sqrt{2}}\)
  9. \(\frac{5\sqrt{72}}{3\sqrt{2}}\)
  10. \(\frac{5\sqrt{105}}{7\sqrt{3}}\)