LemonmathSelf-study sheet

Grade 9 · Lesson 3.1

Square roots and square-root expressions

Name: Date: Time spent: min

Goals

Find the square roots of a non-negative number, and tell square roots apart from the principal square root.
Use the identity $\sqrt{A^2}=|A|$ to evaluate and simplify.
Find the condition for a square-root expression $\sqrt{A}$ to be defined.

Remember

$x$ is a square root of $a\ge0$ when $x^2=a$ . A positive $a$ has two square roots $\pm\sqrt{a}$ ; $0$ has one; a negative number has none.
$\sqrt{a}$ is the principal square root: $\sqrt{a}\ge0$ and $\left(\sqrt{a}\right)^2=a$ .
For $a$ , $b\ge0$ : $a<b\iff\sqrt{a}<\sqrt{b}$ .
$\sqrt{A^2}=|A|$ : it equals $A$ when $A\ge0$ and $-A$ when $A<0$ .
$\sqrt{A}$ is defined $\iff A\ge0$ .

Exercises

Exercise 1

Find the square roots of: a) $121$ ; b) $0.49$ ; c) $\dfrac{25}{36}$ .

Exercise 2

Evaluate: a) $\sqrt{0.64}+\sqrt{1.21}$ ; b) $\left(\sqrt7\right)^2-\sqrt{(-6)^2}$ .

Exercise 3

Compare: a) $6$ and $\sqrt{35}$ ; b) $\sqrt{10}-1$ and $2$ .

Exercise 4

Find the condition on $x$ for each expression to be defined: a) $\sqrt{3x-12}$ ; b) $\sqrt{7-2x}$ ; c) $\sqrt{x^2+4}$ .

Exercise 5

Simplify the following expressions: a) $\sqrt{\left(\sqrt3-1\right)^2}$ ; b) $\sqrt{\left(3-\sqrt{11}\right)^2}$ .

Exercise 6

A square rug has an area of $30$ m². Find its perimeter (rounded to one decimal place).

Exercise 7

Use the formula $t=\sqrt{\dfrac{h}{5}}$ (seconds) for an object falling freely from a height of $h$ metres.

a) A ball is dropped from a height of $80$ m. How long does it take to hit the ground?

b) A marble takes exactly $2.5$ seconds to fall. From what height was it dropped?

Exercise 8

Simplify the expression $Q=\sqrt{x^2-6x+9}-x$ for $x\ge3$ .

Exercise 9

Consider the expression $A=\sqrt{4x-8}$ .

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

b) Find the value of $A$ when $x=6$ .

c) Find $x$ such that $A=6$ .

Exercise 10 Challenge

Find $x$ given that $\sqrt{x^2}=2x-3$ .

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. The principal square root of

49

7

.B.

-7

.C.

7

and

-7

.D.

24.5

Question 2.

\sqrt{(-8)^2}

equals

-8

.B.

8

.C.

64

.D.

\pm8

Question 3. The expression

\sqrt{6-3x}

is defined when

x\ge2

.B.

x\le2

.C.

x\ge-2

.D.

x\le-2

Question 4.

\sqrt{\left(\sqrt5-3\right)^2}

equals

\sqrt5-3

.B.

3-\sqrt5

.C.

3+\sqrt5

.D.

14-6\sqrt5

Part II. True or false

Question 1. Consider the expression

A=\sqrt{2x+10}

. Decide whether each statement is true or false.

a) $A$ is defined when $x\ge-5$ .	T / F
b) When $x=3$ , $A=4$ .	T / F
c) When $x=-3$ , $A=-2$ .	T / F
d) $A=0$ when $x=5$ .	T / F

Part III. Short answer

Question 1. Evaluate

\sqrt{0.81}+\sqrt{(-1.2)^2}

Question 2. The time

t

(seconds) for an object falling freely from a height of

h

metres to hit the ground is given by

t=\sqrt{\dfrac{h}{5}}

. An object falls freely from a height of

61.25

m. After how many seconds does it hit the ground?

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

Hints (only when stuck)

Exercise 1.

What you need: a positive number $a$ has two square roots, $\sqrt{a}$ and $-\sqrt{a}$ .

Steps:

Find the positive number whose square is the given number.
Add its opposite.

Exercise 2.

What you need: $\left(\sqrt{a}\right)^2=a$ for $a\ge0$ ; $\sqrt{a^2}=|a|$ .

Steps:

Take each square root.
Add or subtract.

Exercise 3.

What you need: for $a$ , $b\ge0$ : $a<b\iff\sqrt{a}<\sqrt{b}$ .

Steps:

Write the whole number as a square root: $6=\sqrt{36}$ , $3=\sqrt9$ .
Compare the numbers under the root signs.

Exercise 4.

What you need: $\sqrt{A}$ is defined when $A\ge0$ .

Steps:

Write the inequality $A\ge0$ .
Solve it; remember to reverse the sign when dividing by a negative number.

Exercise 5.

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

Steps:

Find the sign of the number inside (compare $\sqrt3$ with $1$ , and $\sqrt{11}$ with $3$ ).
Remove the absolute value.

Exercise 6.

What you need: a square of area $S$ has side $\sqrt{S}$ .

Steps:

The side is $\sqrt{30}$ m.
The perimeter is $4$ times the side; use the calculator and round only at the end.

Exercise 7.

What you need: $t=\sqrt{\dfrac{h}{5}}\iff t^2=\dfrac{h}{5}\iff h=5t^2$ (for $t\ge0$ ).

Steps:

Part a: substitute $h=80$ .
Part b: use $h=5t^2$ .

Exercise 8.

What you need: the identity $(a-b)^2=a^2-2ab+b^2$ (grade 8) and $\sqrt{A^2}=|A|$ .

Steps:

Write $x^2-6x+9=(x-3)^2$ .
For $x\ge3$ , $|x-3|=x-3$ .

Exercise 9.

What you need: $\sqrt{B}$ is defined when $B\ge0$ ; for $c\ge0$ , $\sqrt{B}=c\iff B=c^2$ .

Steps:

Solve $4x-8\ge0$ .
Substitute $x=6$ .
Solve $4x-8=36$ and check the condition.

Exercise 10.

What you need: $\sqrt{x^2}=|x|\ge0$ , so the right side must also be non-negative.

Steps:

Condition: $2x-3\ge0$ , that is, $x\ge1.5$ .
Then $x>0$ , so $|x|=x$ ; solve $x=2x-3$ .

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