LemonmathSelf-study sheet

Grade 9 · Lesson 3.2

Square roots with multiplication and division

Name: Date: Time spent: min

Goals

Take the square root of a product, and multiply square roots.
Take the square root of a quotient, and divide one square root by another.
Apply both rules to square-root expressions with letters and to real-life problems.

Remember

$\sqrt{ab}=\sqrt a\cdot\sqrt b$ for $a\ge0$ , $b\ge0$ : take the root of each factor, or multiply roots into one root.
$\sqrt{\dfrac ab}=\dfrac{\sqrt a}{\sqrt b}$ for $a\ge0$ , $b>0$ : take the root of the numerator and the denominator, or divide roots into one root.
Both rules hold for non-negative expressions $A$ , $B$ too (with a positive denominator).
Tip: regroup the factors to get perfect squares, as in $810\cdot40=81\cdot400$ .
There is no rule like this for addition or subtraction.

Exercises

Exercise 1

Evaluate: a) $\sqrt{16\cdot0.25}$ ; b) $\sqrt{2.5\cdot40}$ ; c) $\sqrt{12\cdot75}$ .

Exercise 2

Evaluate: a) $\sqrt3\cdot\sqrt{27}$ ; b) $\sqrt{0.2}\cdot\sqrt{180}$ ; c) $\sqrt{1.6}\cdot\sqrt{250}$ .

Exercise 3

Evaluate: a) $\sqrt{\dfrac{49}{81}}$ ; b) $\sqrt{2\dfrac79}$ .

Exercise 4

Evaluate: a) $\dfrac{\sqrt{72}}{\sqrt2}$ ; b) $\dfrac{\sqrt{6.4}}{\sqrt{0.1}}$ ; c) $\dfrac{\sqrt{12.5}}{\sqrt{0.5}}$ .

Exercise 5

Simplify the following expressions: a) $\sqrt{5x}\cdot\sqrt{20x}$ for $x\ge0$ ; b) $\dfrac{\sqrt{27y^3}}{\sqrt{3y}}$ for $y>0$ .

Exercise 6

Simplify the expression $\sqrt{4a^2}-3a$ for $a<0$ .

Exercise 7

A square photo has an area of $150$ cm². It is enlarged into a square with $6$ times the area.

a) How long, in centimetres, is a side of the new photo?

b) How many times as long as the old side is the new side (rounded to two decimal places)?

Exercise 8

The speed (m/s) of a tsunami in water $d$ metres deep is about $v=\sqrt{10d}$ .

a) Find the speed of the wave where the depth is $2.5$ m.

b) Find the speed of the wave where the depth is $4000$ m.

c) At the speed in part b, how many kilometres does the wave travel in $1$ hour?

Exercise 9

Find $x\ge0$ given that $\sqrt{2x}\cdot\sqrt8=12$ .

Exercise 10 Challenge

Without a calculator, evaluate $A=\sqrt{3+\sqrt5}\cdot\sqrt{3-\sqrt5}$ .

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{0.4\cdot90}

equals

6

.B.

36

.C.

0.6

.D.

60

Question 2.

\sqrt{12}\cdot\sqrt3

equals

6

.B.

\sqrt{15}

.C.

36

.D.

4

Question 3. The identity

\sqrt{\dfrac ab}=\dfrac{\sqrt a}{\sqrt b}

holds when

a\ge0

b\ge0

.B.

a\ge0

b>0

.C.

a>0

b\ge0

.D. for all

a

b

Question 4. For

x<0

\sqrt{16x^2}

equals

4x

.B.

-4x

.C.

16x

.D.

-16x

Part II. True or false

Question 1. Let

A=\sqrt{50}\cdot\sqrt2

and

B=\dfrac{\sqrt{98}}{\sqrt2}

. Decide whether each statement is true or false.

a) $A=10$ .	T / F
b) $B=49$ .	T / F
c) $A\cdot B=70$ .	T / F
d) $A-B=\sqrt{51}$ .	T / F

Part III. Short answer

Question 1. Evaluate

\sqrt{1.21\cdot0.25}

Question 2. In shallow water, the speed

v

(m/s) of a sea wave is about

v=\sqrt{10d}

, where

d

(m) is the depth of the water. Find the speed of the wave (in m/s) where the water is

8.1

m deep.

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

Hints (only when stuck)

Exercise 1.

What you need: $\sqrt{ab}=\sqrt a\cdot\sqrt b$ for $a$ , $b\ge0$ .

Steps:

If each factor is a perfect square, take the root of each factor.
If not, multiply to get a perfect square.

Exercise 2.

What you need: $\sqrt a\cdot\sqrt b=\sqrt{ab}$ .

Steps:

Multiply the two numbers under the root signs.
Take the root of the result.

Exercise 3.

What you need: $\sqrt{\dfrac ab}=\dfrac{\sqrt a}{\sqrt b}$ for $a\ge0$ , $b>0$ .

Steps:

Change the mixed number into a fraction.
Take the root of the numerator and of the denominator.

Exercise 4.

What you need: $\dfrac{\sqrt a}{\sqrt b}=\sqrt{\dfrac ab}$ .

Steps:

Divide the two numbers under the root signs.
Take the root of the quotient.

Exercise 5.

What you need: combine into one root, then use $\sqrt{A^2}=|A|$ .

Steps:

Multiply (or divide) the expressions under the root signs.
Write the result as a square and remove the root using the condition.

Exercise 6.

What you need: $\sqrt{4a^2}=\sqrt4\cdot\sqrt{a^2}=2|a|$ (see square roots and square-root expressions).

Steps:

Write $\sqrt{4a^2}=2|a|$ .
For $a<0$ , $|a|=-a$ .

Exercise 7.

What you need: a square of area $S$ has side $\sqrt S$ ; $\dfrac{\sqrt{6S}}{\sqrt S}=\sqrt6$ .

Steps:

The new area is $150\cdot6$ ; take its root.
Divide the two sides using the rule for dividing roots.

Exercise 8.

What you need: $\sqrt{ab}=\sqrt a\cdot\sqrt b$ ; distance equals speed times time.

Steps:

Multiply under the root sign: $10\cdot2.5$ and $10\cdot4000$ .
Convert $1$ hour $=3600$ seconds.

Exercise 9.

What you need: $\sqrt{2x}\cdot\sqrt8=\sqrt{16x}=4\sqrt x$ .

Steps:

Combine the left side into $4\sqrt x$ .
Find $\sqrt x$ , then $x$ .

Exercise 10.

What you need: $\sqrt a\cdot\sqrt b=\sqrt{ab}$ when $a$ , $b\ge0$ ; the identity $(m+n)(m-n)=m^2-n^2$ .

Steps:

Check that $3-\sqrt5\ge0$ .
Combine into one root and use the identity.

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