Surds

Lesson

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

In this example of $-3x+5x$−3`x`+5`x` 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:

$-3x+5x=2x$−3`x`+5`x`=2`x`

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

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.

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

As before, we can view the variable part $\sqrt{x^2y}$√`x`2`y` 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√`x`2`y`+50√`x`2`y`=−31√`x`2`y`

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

However $5\sqrt{x^2y}$5√`x`2`y` can be rewritten in the form $5x\sqrt{y}$5`x`√`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}$6`x`√`y`−√`x``y`

Simplify the expression $11\sqrt{a}-\sqrt{9a}$11√`a`−√9`a`.

Simplify the expression $\sqrt[3]{512v}-5\sqrt[3]{v}$^{3}√512`v`−5^{3}√`v`.

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