Question 11 Mark
Multiply : $\sqrt{2}(\sqrt{8}+\sqrt{18})$
View full question & answer→Question 21 Mark
Divide the surds : $\sqrt{125} \div \sqrt{5}$
Answer$\frac{\sqrt{125}}{\sqrt{5}}=\sqrt{\frac{125}{5}}=\sqrt{25}=5$
(5 is a rational number.)
View full question & answer→Question 31 Mark
Multiply the surds $\sqrt{7} \times \sqrt{42}$
Answer$\sqrt{7} \times \sqrt{42}=\sqrt{7 \times 42}=\sqrt{7 \times 7 \times 6}=7 \sqrt{6}$
( $7 \sqrt{6}$ is an irrational number.)
View full question & answer→Question 41 Mark
Simplify : $8 \sqrt{5}+\sqrt{20}-\sqrt{125}$
Answer$
\begin{aligned}
8 \sqrt{5}+\sqrt{20}-\sqrt{125} & =8 \sqrt{5}+\sqrt{4 \times 5}-\sqrt{25 \times 5} \\
= & 8 \sqrt{5}+2 \sqrt{5}-5 \sqrt{5} \\
= & (8+2-5) \sqrt{5} \\
= & 5 \sqrt{5}
\end{aligned}
$
View full question & answer→Question 51 Mark
Simplify : $13 \sqrt{8}+\frac{1}{2} \sqrt{8}-5 \sqrt{8}$
Answer$=\left(13+\frac{1}{2}-5\right) \sqrt{8}=\left(\frac{26+1-10}{2}\right) \sqrt{8}$$
\begin{aligned}
& =\frac{17}{2} \sqrt{8}=\frac{17}{2} \sqrt{4 \times 2} \\
& =\frac{17}{2} \times 2 \sqrt{2}=17 \sqrt{2}
\end{aligned}
$
View full question & answer→Question 61 Mark
Simplify : $7 \sqrt{3}-29 \sqrt{3}$
Answer$7 \sqrt{3}-29 \sqrt{3}=(7-29) \sqrt{3}=-22 \sqrt{3}$
View full question & answer→Question 71 Mark
Simplify : $7 \sqrt{3}+29 \sqrt{3}$
Answer$7 \sqrt{3}+29 \sqrt{3}=(7+29) \sqrt{3}=36 \sqrt{3}$
View full question & answer→Question 81 Mark
Express the recurring decimal 7.529529529 ... in $\frac{p}{q}$ form
AnswerLet $x=7.529529 \ldots=7 . \overline{529}$
$
\begin{aligned}
& \therefore \quad 1000 x=7529.529529 \ldots=7529 . \overline{529} \\
& \therefore \quad 1000 x-x=7529 . \overline{529}-7 . \overline{529} \\
& \therefore \quad 999 x=7522.0 \quad \therefore \quad x=\frac{7522}{999} \\
& \therefore \quad 7 . \overline{529}=\frac{7522}{999}
\end{aligned}
$
View full question & answer→Question 91 Mark
Express the recurring decimal $0.777 \ldots$ in $\frac{p}{q}$ form.
Answer$
\text { Let } \begin{gathered}
x=0.777 \ldots=0 . \dot{7} \\
\therefore 10 x=7.777 \ldots=7.7 \\
\therefore \quad 10 x-x=7.7-0 . \dot{7} \\
\therefore 9 x=7 \\
\therefore x=\frac{7}{9} \\
\therefore 0.777 \ldots=\frac{7}{9}
\end{gathered}
$
View full question & answer→Question 101 Mark
Divide and write form : $\sqrt{310} \div \sqrt{5}$
Answer$\frac{\sqrt{310}}{\sqrt{5}}=\sqrt{\frac{310}{5}}=\sqrt{\frac{5 \times 62}{5}}=\sqrt{62}$
View full question & answer→Question 111 Mark
Divide and write form : $\sqrt{54} \div \sqrt{27}$
Answer$\frac{\sqrt{54}}{\sqrt{27}}=\sqrt{\frac{54}{27}}=\sqrt{2}$
View full question & answer→Question 121 Mark
Divide and write form : $\sqrt{125} \div \sqrt{50}$
Answer$\frac{\sqrt{125}}{\sqrt{50}}=\sqrt{\frac{125}{50}}=\sqrt{\frac{25 \times 5}{25 \times 2}}=\sqrt{\frac{5}{2}}$
View full question & answer→Question 131 Mark
Divide and write form : $\quad \sqrt{98} \div \sqrt{2}$
Answer$\quad \frac{\sqrt{98}}{\sqrt{2}}=\sqrt{\frac{98}{2}}=\sqrt{49}=7$
View full question & answer→Question 141 Mark
Multiply and write the answer in the simplest form : $\quad 5 \sqrt{8} \times 2 \sqrt{8}$
Answer$\begin{aligned} 5 \sqrt{8} \times 2 \sqrt{8} & =5 \times 2 \times \sqrt{8} \times \sqrt{8} \\ & =5 \times 2 \times 8 \\ \therefore \quad 5 \sqrt{8} \times 2 \sqrt{8} & =80\end{aligned}$
View full question & answer→Question 151 Mark
Multiply and write the answer in the simplest form : $3 \sqrt{8} \times \sqrt{5}$
Answer$\begin{aligned} 3 \sqrt{8} \times \sqrt{5} & =3 \times \sqrt{4 \times 2} \times \sqrt{5} \\ & =3 \times 2 \sqrt{2} \times \sqrt{5} \\ & =6 \sqrt{10} \\ \therefore \quad 3 \sqrt{8} \times \sqrt{5} & =6 \sqrt{10}\end{aligned}$
View full question & answer→Question 161 Mark
Multiply and write the answer in the simplest form : $3 \sqrt{12} \times 7 \sqrt{15}$
Answer$\begin{aligned} 3 \sqrt{12} \times 7 \sqrt{15} & =3 \times \sqrt{4 \times 3} \times 7 \times \sqrt{5 \times 3} \\ & =3 \times 2 \sqrt{3} \times 7 \sqrt{5} \times \sqrt{3} \\ & =3 \times 2 \times 7 \times \sqrt{3} \times \sqrt{3} \times \sqrt{5} \\ & =42 \times 3 \times \sqrt{5} \\ & =126 \sqrt{5} \\ \therefore \quad 3 \sqrt{12} \times 7 \sqrt{15} & = 1 2 6 \sqrt{5}\end{aligned}$
View full question & answer→Question 171 Mark
Multiply and write the answer in the simplest form : $\quad 3 \sqrt{12} \times \sqrt{18}$
Answer$\quad \begin{aligned} 3 \sqrt{12} \times \sqrt{18} & =3 \times \sqrt{4 \times 3} \times \sqrt{9 \times 2} \\ & =3 \times 2 \sqrt{3} \times 3 \sqrt{2} \\ & =3 \times 2 \times 3 \times \sqrt{3} \times \sqrt{2} \\ & =18 \sqrt{6} \\ \therefore \quad 3 \sqrt{12} \times \sqrt{18} & =18 \sqrt{6}\end{aligned}$
View full question & answer→Question 181 Mark
Simplify : $\quad 5 \sqrt{3}+8 \sqrt{3}$
Answer$\begin{array}{ll} & 5 \sqrt{3}+8 \sqrt{3}=(5+8) \sqrt{3}=13 \sqrt{3} \\ \therefore & 5 \sqrt{3}+8 \sqrt{3}=13 \sqrt{3}\end{array}$
View full question & answer→Question 191 Mark
Simplify the following surds : $\quad \sqrt{168}$
Answer$\quad \begin{aligned} \sqrt{168} & =\sqrt{4 \times 42} \\ & =\sqrt{4} \times \sqrt{42} \\ & =2 \sqrt{42}\end{aligned}$
View full question & answer→Question 201 Mark
Simplify the following surds : $\sqrt{112}$
Answer$\quad \begin{aligned} \sqrt{112} & =\sqrt{16 \times 7} \\ & =\sqrt{16} \times \sqrt{7} \\ & =4 \sqrt{7}\end{aligned}$
View full question & answer→Question 211 Mark
Simplify the following surds : $\sqrt{250}$
Answer$\quad \begin{aligned} \sqrt{250} & =\sqrt{25 \times 10} \\ & =\sqrt{25} \times \sqrt{10} \\ & =5 \sqrt{10}\end{aligned}$
View full question & answer→Question 221 Mark
Simplify the following surds : $\sqrt{50}$
Answer$\quad \begin{aligned} \sqrt{50} & =\sqrt{25 \times 2} \\ & =\sqrt{25} \times \sqrt{2} \\ & =5 \sqrt{2}\end{aligned}$
View full question & answer→Question 231 Mark
Simplify the following surds : $\sqrt{27}$
Answer$\quad \begin{aligned} \sqrt{27} & =\sqrt{9 \times 3} \\ & =\sqrt{9} \times \sqrt{3} \\ & =3 \sqrt{3}\end{aligned}$
View full question & answer→Question 241 Mark
State which of the following are surds Justify. : $\sqrt{\frac{22}{7}}$
Answer$\sqrt{\frac{22}{7}}$ is a surd because $\frac{22}{7}$ is a positive rational number, 2 is a positive integer greater than 1 and $\sqrt{\frac{22}{7}}$ is irrational.
View full question & answer→Question 251 Mark
State which of the following are surds Justify. : $\sqrt[3]{64}$
Answer$\sqrt[3]{64}$ is not a surd because
$
\begin{aligned}
\sqrt[3]{64} & =(64)^{\frac{1}{3}} \\
& =\left(4^3\right)^{\frac{1}{3}}
\end{aligned}
$
$=4$, which is not an irrational number.
View full question & answer→Question 261 Mark
State which of the following are surds Justify. : $\sqrt{256}$
Answer$\sqrt{256}$ is not a surd because
$
\begin{aligned}
\sqrt{256} & =(256)^{\frac{1}{2}} \\
& =\left(16^2\right)^{\frac{1}{2}}
\end{aligned}
$
$=16$, which is not an irrational number.
View full question & answer→Question 271 Mark
State which of the following are surds Justify. : $\sqrt[5]{81}$
Answer$\sqrt[5]{81}$ is a surd because 81 is a positive rational number, 5 is a positive integer greater than 1 and $\sqrt[5]{81}$ is irrational.
View full question & answer→Question 281 Mark
State which of the following are surds Justify. : $\sqrt[4]{16}$
Answer$\sqrt[4]{16}$ is not a surd because
$
\begin{aligned}
\sqrt[4]{16} & =(16)^{\frac{1}{4}} \\
& =\left(2^4\right)^{\frac{1}{4}}
\end{aligned}
$
$=2$, which is not an irrational number.
View full question & answer→Question 291 Mark
State which of the following are surds Justify. : $\quad \sqrt[3]{51}$
Answer$\sqrt[3]{51}$ is a surd because 51 is a positive rational number, 3 is a positive integer greater than 1 and $\sqrt[3]{51}$ is irrational.
View full question & answer→Question 301 Mark
Write the following rational numbers in decimal form. : $\frac{17}{8}$
Answer$\frac{17}{8}=\frac{17}{8} \times \frac{125}{125}=\frac{2125}{1000}=2.125$
View full question & answer→Question 311 Mark
Write the following rational numbers in decimal form. : $\frac{4}{5}$
Answer$\frac{4}{5} \times \frac{2}{2}=\frac{8}{10}=0.8$
View full question & answer→Question 321 Mark
Write the following rational numbers in decimal form. : $\frac{23}{7}$
Answer$\frac{23}{7}=3.2857142857 \ldots=3 . \overline{285714}$
View full question & answer→Question 331 Mark
Write the following rational numbers in decimal form. : $\frac{25}{99}$
Answer$\frac{25}{99}=\frac{4}{4} \times \frac{25}{99}=\frac{1}{4} \times \frac{100}{99}=\frac{1}{4} \times 1.010101 \ldots=0.2525 \ldots=0 . \overline{25}$.
View full question & answer→Question 341 Mark
Write the following rational numbers in decimal form. : $\quad \frac{127}{200}$
Answer$\frac{127}{200}=\frac{127}{200} \times \frac{5}{5}=\frac{635}{1000}=0.635$
View full question & answer→