# 2.2.1 Squares, Square Roots, Cube and Cube Roots, PT3 Practice 1

2.2.1 Squares, Square Roots, Cube and Cube Roots, PT3 Practice 1

Question 1:
Calculate the values of the following:
$\begin{array}{l}\text{(a)}\sqrt{\frac{50}{98}}\\ \text{(b)}\sqrt{1\frac{17}{64}}\\ \text{(c)}\sqrt{81}-\sqrt{0.01}\\ \text{(d)}\sqrt{3.24}\end{array}$

Solution:
$\text{(a)}\sqrt{\frac{50}{98}}=\sqrt{\frac{\overline{)50}25}{\overline{)98}49}}=\sqrt{\frac{25}{49}}=\frac{5}{7}$

$\text{(b)}\sqrt{1\frac{17}{64}}=\sqrt{\frac{81}{64}}=\frac{9}{8}=1\frac{1}{8}$

$\begin{array}{l}\text{(c)}\sqrt{81}-\sqrt{0.01}=9-\sqrt{\frac{1}{100}}\\ \text{}=9-\frac{1}{10}\\ \text{}=9-0.1\\ \text{}=8.9\end{array}$

$\begin{array}{l}\text{(d)}\sqrt{3.24}=\sqrt{3\frac{24}{100}}=\sqrt{3\frac{6}{25}}\\ \text{}=\sqrt{\frac{81}{25}}\\ \text{}=\frac{9}{5}=1\frac{4}{5}\end{array}$

Question 2:
Calculate the values of the following:
$\begin{array}{l}\text{(a)}\sqrt[3]{\frac{16}{250}}\\ \text{(b)}\sqrt[3]{-\frac{4}{256}}\\ \text{(c)}\sqrt[3]{0.008}\\ \text{(d)}\sqrt[3]{0.729}\end{array}$

Solution:
$\text{(a)}\sqrt[3]{\frac{16}{250}}=\sqrt[3]{\frac{8}{125}}=\frac{2}{5}$

$\text{(b)}\sqrt[3]{-\frac{4}{256}}=\sqrt[3]{-\frac{1}{64}}=-\frac{1}{4}$

$\begin{array}{l}\text{(c)}\sqrt[3]{0.008}=\sqrt[3]{\frac{8}{1000}}\\ \text{=}\frac{2}{10}\\ \text{}=0.2\end{array}$

$\begin{array}{l}\text{(d)}\sqrt[3]{-0.729}=\sqrt[3]{-\frac{729}{1000}}\\ \text{}=-\frac{9}{10}\\ \text{}=-0.9\end{array}$

Question 3:
Find the value of $\sqrt[3]{3\frac{3}{8}}+\sqrt{2\frac{14}{25}}.$

Solution:
$\begin{array}{l}\sqrt[3]{3\frac{3}{8}}+\sqrt{2\frac{14}{25}}=\sqrt[3]{\frac{27}{8}}+\sqrt{\frac{64}{25}}\\ \text{}=\frac{3}{2}+\frac{8}{5}\\ \text{}=\frac{31}{10}=3\frac{1}{10}\end{array}$

Question 4:
Find the values of the following:
(a) 1 – (–0.3)3.
(b) ${\left(2.1÷\sqrt[3]{27}\right)}^{2}$

Solution:
(a)
1 – (–0.3)3 = 1 – [(–0.3) × (–0.3) × (–0.3)]
= 1 – (–0.027)
= 1 + 0.027
= 1.027

(b)
$\begin{array}{l}{\left(2.1÷\sqrt[3]{27}\right)}^{2}={\left(2.1÷3\right)}^{2}\\ \text{}={\left(0.7\right)}^{2}\\ \text{}=0.49\end{array}$

Question 5:
Find the values of the following:
$\begin{array}{l}\text{(a)}{\left(9+\sqrt[3]{-8}\right)}^{2}\\ \text{(b)}\sqrt{144}÷\sqrt[3]{216}×{0.3}^{3}\end{array}$

Solution:
$\begin{array}{l}\text{(a)}{\left(9+\sqrt[3]{-8}\right)}^{2}={\left[9+\left(-2\right)\right]}^{2}\\ \text{}={7}^{2}\\ \text{}=49\end{array}$

$\begin{array}{l}\text{(b)}\sqrt{144}÷\sqrt[3]{216}×{0.3}^{3}\\ \text{}=144÷6×\left(0.3×0.3×0.3\right)\\ \text{}=24×0.027\\ \text{}=0.648\end{array}$