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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:


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



example 1

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

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


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}$81x2y+50x2y.

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:



Worked example

Simplify the expression $5\sqrt{x^2y}+x\sqrt{y}-\sqrt{xy}$5x2y+xyxy 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}$5x2y and $-\sqrt{xy}$xy are not like terms, since their variable parts are different.

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


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

So the resulting simplified expression is:



Practice questions

question 1

Simplify the expression $11\sqrt{a}-\sqrt{9a}$11a9a.


Simplify the expression $\sqrt[3]{512v}-5\sqrt[3]{v}$3512v53v.

question 3

Simplify the expression $\sqrt{ax^5}+x^2\sqrt{ax}$ax5+x2ax, where $x$x represents a positive number.

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