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.
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√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
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
Simplify the expression $11\sqrt{a}-\sqrt{9a}$11√a−√9a.
Simplify the expression $\sqrt[3]{512v}-5\sqrt[3]{v}$3√512v−53√v.
Simplify the expression $\sqrt{ax^5}+x^2\sqrt{ax}$√ax5+x2√ax, where $x$x represents a positive number.