Adding and Subtracting Radical Expressions

Adding and Subtracting Radical Expressions exercise

• Explain how to add and subtract radical expressions.

  Hints

  Only combine like terms.

  You can combine $8x-9x$ as follows: $8x-9x=(8-9)x=-1x=-x$.

  Remember to keep track of which letters were assigned to which square roots, so that you can re-replace the variables with the right square roots later.

  Solution

  There are two different numbers under square roots, $6$ and $7$, so we want to assign two different letters for each root:

  • $\sqrt6=x$
  • $\sqrt7=y$

  Now replace all roots with the same variable in the expression $8\sqrt6+3\sqrt7-9\sqrt6+2\sqrt7$ to get

  $8x+3y-9x+2y$.

  This equation can be solved with the same rules for adding variables:

  $8x+3y-9x+2y=8x-9x+3y+2y=-x+5y$.

  Lastly re-replace the variables with the original square roots to get the result:

  $8\sqrt6+3\sqrt7-9\sqrt6+2\sqrt7=-\sqrt6+5\sqrt7$.

• Simplify the following expression.

  Hints

  You can only combine like terms. For instance, we have $3x+3y-2x=3x-2x+3y=x-3y$.

  Here is an example of adding two radical expressions together.

  $\begin{array}{rcllll} 4\sqrt8+5\sqrt8&=&~?&|& \text{assign } \sqrt8=x\\ 4x+5x&=&~9x&|&\text{re-replace } x=\sqrt8\\ 4\sqrt8+5\sqrt8&=&~9\sqrt8 \end{array}$

  Solution

  There are two different numbers under square roots, $3$ and $5$. For these square roots we assign two different letters:

  • $\sqrt3=x$
  • $\sqrt5=y$

  Next we replace all roots with the same variable in the expression $4\sqrt3+5\sqrt5+6\sqrt3-2\sqrt5$ to get $4x+5y+6x-2y$.

  We solve this equation with the same rules for adding or subtracting variables:

  $4x+5y+6x-2y=4x+6x+5y-2y=10x+3y$.

  Lastly we re-replace the variables with the original square roots to get

  $4\sqrt3+5\sqrt5+6\sqrt3-2\sqrt5=10\sqrt3+3\sqrt5$.

• Decide which terms can be added or subtacted.

  Hints

  Whether or not two radical expressions could be added or subtracted depends only on the numbers under the square roots.

  You can only add or subtract radical expressions with same number under the square roots.

  For example, you could add or subtract $3\sqrt5$ and $4\sqrt5$, but you can't add or subtract $3\sqrt5$ and $4\sqrt4$.

  Solution

  Keep in mind that you can only add or subtract radical expressions with the same number under the square roots.

  So you can add or subtract the following expressions:

  • $3\sqrt6$ and $-\sqrt6$ as well as $7\sqrt6$
  • $2\sqrt3$ and $\sqrt3$ as well as $6\sqrt3$
  • $\sqrt7$ and $6\sqrt7$ as well as $3\sqrt7$

• Find the errors in the calculation.

  Hints

  First count the different numbers under square roots.

  Keep in mind that you only can combine radical expressions with the same number under the square roots.

  There are $8$ errors in total.

  Solution

  Here you see three different numbers under square roots. We assign the variables

  • $\sqrt3=x$
  • $\sqrt5=y$
  • $\sqrt2=z$

  and replace the square roots to get

  $3x+4y-2z+5y-3z+2x$.

  We handle this expression by combining like terms:

  $3x+4y-2z+5y-3z+2x=3x+2x+4y+5y-2z-3z=5x+9y-5z$.

  Lastly we re-replace the variables with the corresponding square roots to get the desired solution:

  $3\sqrt3+4\sqrt5-2\sqrt2+5\sqrt5-3\sqrt2+2\sqrt3=5\sqrt3+9\sqrt5-5\sqrt2$.

• Determine when radical expressions can be added or subtracted.

  Hints

  You can only combine like terms. For example:

  • $3x+4x=7x$
  • $3x+4y$ can't be simplified.

  You can't simplify an expression like $3\sqrt3+4\sqrt4$.

  Remember Maxine's hint on the tape: "Stop combining things that shouldn't be combined.“

  Solution

  What radical expressions can be added or subtracted?

  • Could $3\sqrt3$ and $2\sqrt4$ be added or subtracted? No!
  • Could $3\sqrt3$ and $2\sqrt3$ be added or subtracted? Yes!

  What's the difference?

  In the first example the numbers under the square roots are different, while in the second both are the same.

  You can only add or subtract radical expressions if the numbers under the square roots are the same.

  Let's simplify the example above:

  • $3\sqrt3+2\sqrt3=5\sqrt3$
  • $3\sqrt3-2\sqrt3=\sqrt3$

• Add or subtract the following radical expressions.

  Hints

  Just follow those steps.

  You can only combine radical expressions with the same numbers under the square roots.

  Solution

  For adding or subtracting radical expression just follow the following steps:

  1. Assign a different letter for each root in the equation.
  2. Replace all instances of the root with the same variable.
  3. Solve the equation with the same rules for adding variables.
  4. Re-replace the variables with the original square roots.

  Let's practice this with a few examples:

  $~$

  $-3\sqrt5+4\sqrt7+4\sqrt5+3\sqrt7$

  1. $\sqrt5=x$ and $\sqrt 7=y$
  2. $-3x+4y+4x+3y$
  3. $-3x+4y+4x+3y=-3x+4x+4y+3y=x+7y$
  4. $-3\sqrt5+4\sqrt7+4\sqrt5+3\sqrt7=\sqrt5+7\sqrt7$

  $~$

  $5\sqrt7-4\sqrt5+4\sqrt7+3\sqrt5$

  1. $\sqrt7=x$ and $\sqrt 5=y$
  2. $5x-4y+4x+3y$
  3. $5x-4y+4x+3y=5x+4x-4y+3y=9x-y$
  4. $5\sqrt7-4\sqrt5+4\sqrt7+3\sqrt5=9\sqrt7-\sqrt5$

  $~$

  $\sqrt7+6\sqrt5+2\sqrt7-2\sqrt5$

  1. $\sqrt7=x$ and $\sqrt5=y$
  2. $x+6y+2x-2y$
  3. $x+6y+2x-2y=x+2x+6y-2y=3x+4y$
  4. $\sqrt7+6\sqrt5+2\sqrt7-2\sqrt5=3\sqrt7+4\sqrt5$

  $~$

  $7\sqrt7-2\sqrt5+4\sqrt5-3\sqrt7$

  1. $\sqrt7=x$ and $\sqrt 5=y$
  2. $7x-2y+4y-3x$
  3. $7x-2y+4y-3x=7x-3x-2y+4y=4x+2y$
  4. $7\sqrt7-2\sqrt5+4\sqrt5-3\sqrt7=4\sqrt7+2\sqrt5$