Add and subtract algebraic surds

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

In this example of $-3x+5x$−3x+5x each term has a variable part of $x$x, but different constants multiplied in front (which are $-3$−3 and $5$5).

We can combine these like terms by adding their coefficients:

$-3x+5x=2x$−3x+5x=2x

In general the variable part doesn't have to be $x$x, and could be any algebraic expression including surds.

 

Exploration

example 1

Consider the expression $6\sqrt{y}+5\sqrt{y}$6√y+5√y.

Each term in the expression has the variable part $\sqrt{y}$√y and differ by the constants multiplied out front, which are $6$6 and $5$5.

So, the terms $6\sqrt{y}$6√y and $5\sqrt{y}$5√y are like terms and we can combine them by adding their coefficients:

$6\sqrt{y}+5\sqrt{y}=11\sqrt{y}$6√y+5√y=11√y

In this way we can view $\sqrt{y}$√y as we would with any other variable part, since after all it's just a placeholder for a number we do not know.

 

example 2

Let's now consider a more involved expression, such as $-81\sqrt{x^2y}+50\sqrt{x^2y}$−81√x2y+50√x2y.

As before, we can view the variable part $\sqrt{x^2y}$√x2y as just another variable representing a number. In this way the two terms still have the same variable part, and so they are like terms.

So we can combine the coefficients just like we did before:

$-81\sqrt{x^2y}+50\sqrt{x^2y}=-31\sqrt{x^2y}$−81√x2y+50√x2y=−31√x2y

 

Worked example

Simplify the expression $5\sqrt{x^2y}+x\sqrt{y}-\sqrt{xy}$5√x2y+x√y−√xy where $x$x and $y$y are positive.

Think: We first want to identify which terms are like terms.

We know that $5\sqrt{x^2y}$5√x2y and $-\sqrt{xy}$−√xy are not like terms, since their variable parts are different.

However $5\sqrt{x^2y}$5√x2y can be rewritten in the form $5x\sqrt{y}$5x√y, and so it is a like term with $x\sqrt{y}$x√y.

Do:

$5\sqrt{x^2y}+x\sqrt{y}-\sqrt{xy}$5√x2y+x√y−√xy	$=$=	$5x\sqrt{y}+x\sqrt{y}-\sqrt{xy}$5x√y+x√y−√xy	(Since $\sqrt{x^2}=x$√x2=x for positive $x$x)
	$=$=	$6x\sqrt{y}-\sqrt{xy}$6x√y−√xy	(Combining like terms)

So the resulting simplified expression is:

$6x\sqrt{y}-\sqrt{xy}$6x√y−√xy

 

Practice questions

question 1

QUESTION 2

question 3