LemonmathSelf-study sheet

Grade 9 · Lesson 3.3

Transforming and simplifying expressions with square roots

Name: Date: Time spent: min

Goals

Take a factor outside the root sign and move a factor inside it, to calculate and to compare.
Remove the denominator under a root sign and rationalise a denominator.
Simplify expressions with square roots, step by step, as in the grade 10 entrance exam.

Remember

$\sqrt{A^2B}=|A|\sqrt B$ ( $B\ge0$ ): split off a square and take it outside.
$A\sqrt B=\sqrt{A^2B}$ when $A\ge0$ ; $A\sqrt B=-\sqrt{A^2B}$ when $A<0$ . Useful for comparing.
Like surds: $a\sqrt c+b\sqrt c=(a+b)\sqrt c$ .
$\dfrac{A}{\sqrt B}=\dfrac{A\sqrt B}{B}$ ; with a denominator $\sqrt A\pm B$ or $\sqrt A\pm\sqrt B$ , multiply the numerator and the denominator by the conjugate.
Simplifying: conditions → factorise → common denominator → collect terms → check with one value.

Exercises

Exercise 1

Take the factor outside the root sign: a) $\sqrt{75}$ ; b) $\sqrt{200}$ ; c) $\sqrt{28a^2}$ for $a<0$ .

Exercise 2

Evaluate $\sqrt{20}-\sqrt{45}+3\sqrt5$ .

Exercise 3

a) Compare $4\sqrt3$ and $3\sqrt5$ .

b) Put in increasing order: $2\sqrt7$ , $3\sqrt3$ , $4\sqrt2$ .

Exercise 4

Rationalise the denominator: a) $\dfrac{10}{\sqrt5}$ ; b) $\dfrac{2}{\sqrt3+1}$ ; c) $\dfrac{4}{\sqrt7-\sqrt3}$ .

Exercise 5

Remove the denominator under the root sign: a) $\sqrt{\dfrac23}$ ; b) $\sqrt{\dfrac58}$ .

Exercise 6

Evaluate $\left(\sqrt{12}+\sqrt{27}\right)\cdot\sqrt3$ .

Exercise 7

Three square garden beds with areas of $18$ m², $32$ m² and $50$ m² are placed side by side in a row. Find the total length of their three sides along the row: write it in a neat form, then round to two decimal places.

Exercise 8

Simplify the expression $B=\dfrac{a\sqrt b+b\sqrt a}{\sqrt a+\sqrt b}$ for $a>0$ , $b>0$ .

Exercise 9

Consider the expression $P=\dfrac{\sqrt x}{\sqrt x-3}-\dfrac{3}{\sqrt x+3}$ .

a) Find the condition for $P$ to be defined.

b) Simplify $P$ .

c) Find the value of $P$ when $x=36$ .

d) Find $x$ such that $P=2$ .

Exercise 10 Challenge

Evaluate $C=\sqrt{4-2\sqrt3}+\sqrt{4+2\sqrt3}$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1.

\sqrt{72}

equals

6\sqrt2

.B.

36\sqrt2

.C.

2\sqrt6

.D.

3\sqrt2

Question 2. Rationalising the denominator of

\dfrac{6}{\sqrt2}

gives

3\sqrt2

.B.

6\sqrt2

.C.

\sqrt3

.D.

3

Question 3.

\dfrac{1}{\sqrt5-2}

equals

\sqrt5+2

.B.

\sqrt5-2

.C.

\dfrac{\sqrt5+2}{3}

.D.

-\sqrt5-2

Question 4. For

a<0

a\sqrt3

equals

\sqrt{3a^2}

.B.

-\sqrt{3a^2}

.C.

\sqrt{3a}

.D.

-\sqrt{3a}

Part II. True or false

Question 1. Let

A=\sqrt{50}-\sqrt{18}+\sqrt8

. Decide whether each statement is true or false.

a) $\sqrt{50}=5\sqrt2$ .	T / F
b) $A=\sqrt{40}$ .	T / F
c) $A\cdot\sqrt2=8$ .	T / F
d) $\dfrac1A=\dfrac{\sqrt2}{8}$ .	T / F

Part III. Short answer

Question 1. Evaluate

\dfrac{\sqrt{75}-\sqrt{12}}{\sqrt3}

Question 2. A rectangular plot is

\sqrt{98}

m long and

\sqrt{32}

m wide. Find its perimeter in metres (give your answer to 1 decimal place).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $\sqrt{A^2B}=|A|\sqrt B$ .

Steps:

Find the largest perfect square that divides the number under the root.
Take its root outside; with letters, look at the sign.

Exercise 2.

What you need: take factors outside the root sign to get like surds.

Steps:

Write $\sqrt{20}=2\sqrt5$ , $\sqrt{45}=3\sqrt5$ .
Add and subtract the coefficients.

Exercise 3.

What you need: for $A\ge0$ , $A\sqrt B=\sqrt{A^2B}$ .

Steps:

Move the coefficients inside the root sign.
Compare the numbers under the roots.

Exercise 4.

What you need: $\dfrac A{\sqrt B}=\dfrac{A\sqrt B}B$ ; multiply by the conjugate of the denominator.

Steps:

Choose what to multiply the numerator and the denominator by.
Work out the denominator with the identity, then simplify.

Exercise 5.

What you need: $\sqrt{\dfrac AB}=\dfrac{\sqrt{AB}}{|B|}$ .

Steps:

Multiply the numerator and the denominator under the root so that the denominator is a perfect square.
Take the root of the denominator.

Exercise 6.

What you need: $\sqrt a\cdot\sqrt b=\sqrt{ab}$ (see square roots with multiplication and division).

Steps:

Expand the bracket.
Combine into one root and take the root.

Exercise 7.

What you need: a square of area $S$ has side $\sqrt S$ ; take factors outside the root sign.

Steps:

Write the three sides in the form $a\sqrt2$ .
Add, then use the calculator.

Exercise 8.

What you need: $a=\left(\sqrt a\right)^2$ for $a\ge0$ ; take out a common factor.

Steps:

Write the numerator as $\sqrt a\sqrt b\left(\sqrt a+\sqrt b\right)$ .
Cancel with the denominator.

Exercise 9.

What you need: the common denominator is $\left(\sqrt x-3\right)\left(\sqrt x+3\right)=x-9$ .

Steps:

We need $x\ge0$ and $\sqrt x\neq3$ .
Write over the common denominator and collect the numerator.
Substitute $x=36$ ; solve the equation in d) and check the condition.

Exercise 10.

What you need: $\left(\sqrt3\pm1\right)^2=4\pm2\sqrt3$ and $\sqrt{A^2}=|A|$ .

Steps:

Write $4-2\sqrt3=\left(\sqrt3-1\right)^2$ and $4+2\sqrt3=\left(\sqrt3+1\right)^2$ .
Remove the roots, looking at the sign of $\sqrt3-1$ .

Self-check

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

Redo the ones you missed after: ☐ 1 day ☐ 3 days ☐ 7 days

Scan for step-by-step solutions

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐