Grade 9

Square roots: practice problems

A square root of $a$ is a number whose square is $a$ . Calculations with roots start by writing everything in the form $a\sqrt{b}$ .

Basic, Standard, Advanced: Grade 9 · Term 1

BasicStandardAdvancedMix

0Make a worksheet on this topic

Square roots and the form a√b

A positive number $a$ has two square roots, $\pm\sqrt{a}$ . Take square factors outside the root: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ .

Calculating with roots

Rules

\sqrt{a} \times \sqrt{b} = \sqrt{ab},\qquad \sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}},\qquad m\sqrt{a} + n\sqrt{a} = (m + n)\sqrt{a}

Only like surds can be added: $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$ .

Rationalising and approximations

\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}

With $\sqrt{2} = 1.414$ , $\sqrt{200} = 10\sqrt{2} = 14.14$ . Since $3 < \sqrt{13} < 4$ , the integer part of $\sqrt{13}$ is 3 and its fractional part is $\sqrt{13} - 3$ .

Q1BasicGr 9 · Term 1Square roots

Which list puts the three numbers in increasing order? $4,\ \sqrt{15},\ 3\sqrt{2}$

A $4 < \sqrt{15} < 3\sqrt{2}$
B $3\sqrt{2} < \sqrt{15} < 4$
C $4 < 3\sqrt{2} < \sqrt{15}$
D $\sqrt{15} < 4 < 3\sqrt{2}$

Answer

\text{D}

Solution

As square roots:
$4 = \sqrt{16},\quad \sqrt{15} = \sqrt{15},\quad 3\sqrt{2} = \sqrt{18}$
So
$\sqrt{15} < 4 < 3\sqrt{2}$

Q2StandardGr 9 · Term 1Square roots

Simplify.

\sqrt{2}(\sqrt{6} + 1)

Answer

\sqrt{2} + 2\sqrt{3}

Solution

Expand and simplify.
$\sqrt{2}(\sqrt{6} + 1) = \sqrt{2} + 2\sqrt{3}$

Q3AdvancedGr 9 · Term 1Square roots

Find $x^2 - y^2$ when $x = \sqrt{3} + \sqrt{2}$ and $y = \sqrt{3} - \sqrt{2}$ .

Answer

4\sqrt{6}

Solution

First compute:
$x + y = 2\sqrt{3},\quad x - y = 2\sqrt{2}$
Rewrite and substitute.
$x^2 - y^2 = (x + y)(x - y) = 2\sqrt{3} \times 2\sqrt{2} = 4\sqrt{6}$

Q1BasicGr 9 · Term 1Square roots

Which list puts the three numbers in increasing order? $2,\ \sqrt{2},\ 2\sqrt{2}$

A $2 < \sqrt{2} < 2\sqrt{2}$
B $\sqrt{2} < 2\sqrt{2} < 2$
C $2\sqrt{2} < 2 < \sqrt{2}$
D $\sqrt{2} < 2 < 2\sqrt{2}$

Q2StandardGr 9 · Term 1Square roots

Simplify.

(\sqrt{7} - \sqrt{2})^2

Q3AdvancedGr 9 · Term 1Square roots

How many natural numbers $a$ satisfy $6 < \sqrt{a} < 7$ ?

Teleport

Main features

On this page

Topics

Display

Square roots: practice problems

Square roots and the form a√b

Calculating with roots

Rationalising and approximations

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Math problem generator

Square roots and the form a√b

Calculating with roots

Rationalising and approximations

Worked examples

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Practice problems

Hint

Answer

Solution

Hint

Answer

Solution

Hint

Answer

Solution

Related topics

Expanding and factorising

The Pythagorean theorem