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2.02 Adding and subtracting surds

Lesson

Introduction

We know that we can add or subtract algebraic terms, such as -3x + 5x, as long as the terms have the same variable part.

In this example of -3x + 5x each term has a variable part of x, but different coefficients (which are -3 and 5).

We can combine these like terms by adding their coefficients: -3x + 5x = 2x

In general the variable part doesn't actually have to be a pronumeral. Can we use the same rules for adding and subtracting surds?

Add and subtract surds

Consider an expression such as \sqrt{16}+\sqrt{9}. If we evaluate each root we get 4+3=7, but if we try to combine the terms and evaluate the addition under one root sign, that is \sqrt{16+9}, we get \sqrt{25}=5.

So we can see that \,\sqrt{a}+\sqrt{b}\, is not equal to \sqrt{a+b}. The same thing happens to \sqrt{a}-\sqrt{b} as subtraction can be thought of as adding a negative number.

So how can we add and subtract surds?

If a and b are different values, we can not simplify the expression \sqrt{a}+\sqrt{b}.

But if they were the same number, then \,\sqrt{a}+\sqrt{a}=2 \times \sqrt{a} = 2\sqrt{a}\, as whenever we add two of the same things, we can just simplify it to 2 times the thing. This can be extended to adding or subtracting multiples of surds, such as \,5\sqrt{3}-3\sqrt{3}=2\sqrt{3}\, as 5 things take away 3 things equals 2 things.

We can add and subtract like surds:

c\sqrt{a}+d\sqrt{a}=(c+d)\sqrt{a}

c\sqrt{a}-d\sqrt{a}=(c-d)\sqrt{a}

Examples

Example 1

Simplify: 10\sqrt{2}+14\sqrt{2}

Worked Solution
Create a strategy

Add like surds.

Apply the idea

Since both terms have irrational part \sqrt{2}, they are like surds and we can add 10 and 14 together. 10\sqrt{2}+14\sqrt{2} = 24\sqrt{2}

Example 2

Simplify: 6\sqrt{7}+7\sqrt{5}-3\sqrt{7}+8\sqrt{5}

Worked Solution
Create a strategy

Add like surds.

Apply the idea
\displaystyle 6\sqrt{7}+7\sqrt{5}-3\sqrt{7}+8\sqrt{5}\displaystyle =\displaystyle (6\sqrt{7}-3\sqrt{7})+(7\sqrt{5}+8\sqrt{5})Group like surds
\displaystyle =\displaystyle 15\sqrt{5}+3\sqrt{7}Add like surds
Idea summary

We can add and subtract like surds:

c\sqrt{a}+d\sqrt{a}=(c+d)\sqrt{a}

c\sqrt{a}-d\sqrt{a}=(c-d)\sqrt{a}

Simplify surds

Sometimes we are asked to add and subtract surds that do no have the same number under the root sign. If we can simplify the surd so that they have the same number under the root sign, we can then collect like surds.

For example, \sqrt{12}-\sqrt{3} seems to be impossible to simplify any further, as we have two different surds involved.

However we can simplify \sqrt{12}=\sqrt{4 \times 3} down to 2\sqrt{3} so \sqrt{12}-\sqrt{3}=2\sqrt{3}-\sqrt{3} which after collecting like surds, evaluates to \sqrt{3}.

So when dealing with these kinds of problems, make sure to simply all surds first.

Examples

Example 3

Simplify completely: \sqrt{45}+\sqrt{80}

Worked Solution
Create a strategy

Simplify each surd before adding.

Apply the idea

We can use trial and error by dividing 45 and 80 by perfect squares such as 4,\,9,\,16,\,25,\,36,\, \ldots on the calculator.

By doing this, we can find that 45 = 9 \times 5 where 9 is a perfect square, and 80 = 16 \times 5 where 16 is a perfect square.

\displaystyle \sqrt{45}+\sqrt{80}\displaystyle =\displaystyle \sqrt{9} \times \sqrt{5} + \sqrt{16} \times \sqrt{5}Write the surds as a products of their factors
\displaystyle =\displaystyle 3 \sqrt{5} + 4 \sqrt{5}Evaluate \sqrt{9} and \sqrt{16}
\displaystyle =\displaystyle 7\sqrt{5}Add like surds
Idea summary

Before we add and subtract surds, we make sure that all terms are already in their simplest form.

Outcomes

VCMNA355 (10a)

Define rational and irrational numbers and perform operations with surds and fractional indices.

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