Question 12 Marks
By what number should $(-30)^{-1}$ be divided to get $(6)^{-1}$ ?
AnswerThe required number $=(-30)^{-1}\div(6)^{-1}$
$=\frac{-1}{30}\div\frac{1}{6}$
$=\frac{-1}{30}\times\frac{6}{1}$
$=\frac{-1}{5}$
View full question & answer→Question 22 Marks
Express the following in power notation:
$\Big(\frac{-4}{7}\Big)^3$
Answer $\Big(\frac{-4}{7}\Big)^3=\frac{-4}{7}\times\frac{-4}{7}\times\frac{-4}{7}$
$=\frac{(-4)\times(-4)\times(-4)}{7\times7\times7}$
$=\frac{-64}{343}$
View full question & answer→Question 32 Marks
Express the following as a rational number:
$(5^{-1}-7^{-1})^{-1}$
Answer $(5^{-1}-7^{-1})^{-1}=\Big(\frac{1}{5}-\frac{1}{7}\Big)^{-1}$
$=\Big(\frac{7-5}{35}\Big)^{-1}=\Big(\frac{2}{35}\Big)^{-1}$
$=\Big(\frac{35}{2}\Big)^{1}$
$=\frac{35}{2}$
View full question & answer→Question 42 Marks
Express the following in power notation: $\Big(\frac{1}{6}\Big)^3$
Answer$\Big(\frac{1}{6}\Big)^3=\frac{1}{6}\times\frac{1}{6}\times\frac{1}{6}$ $=\frac{1\times1\times1}{6\times6\times6}$ $=\frac{1}{216}$
View full question & answer→Question 52 Marks
Simplify and express each as a rational number: $\Big(\frac{4}{3}\Big)^{-3}\times \Big(\frac{4}{3}\Big)^{-2}$
Answer$\Big(\frac{4}{3}\Big)^{-3}\times \Big(\frac{4}{3}\Big)^{-2}$ $=\Big(\frac{3}{4}\Big)^{-2}\times\Big(\frac{3}{4}\Big)^2$ $=\Big(\frac{3}{4}\Big)^{3+2}=\Big(\frac{3}{4}\Big)^{5}$ $=\frac{3}{4}\times\frac{3}{4}\times\frac{3}{4}\times\frac{3}{4}\times\frac{3}{4}$ $=\frac{243}{1024}$
View full question & answer→Question 62 Marks
Express the following in power notation: $\Big(\frac{-3}{2}\Big)^4$
Answer$\Big(\frac{-3}{2}\Big)^4=\frac{-3}{2}\times\frac{-3}{2}\times\frac{-3}{2}\times\frac{-3}{2}$ $=\frac{(-3)\times(-3)\times(-3)\times(-3)}{2\times2\times2\times2}$ $=\frac{81}{16}$
View full question & answer→Question 72 Marks
Express the following as a rational number:
$\Bigg\{\Big(\frac{3}{2}\Big)\div\Big(\frac{-2}{5}\Big)^{-1}\Bigg\}$
Answer $\Bigg\{\Big(\frac{3}{2}\Big)\div\Big(\frac{-2}{5}\Big)^{-1}\Bigg\}$
$=\Bigg[\Big(\frac{2}{3}\Big)^1\div\Big(\frac{5}{-2}\Big)^1\Bigg]$
$=\frac{2}{3}\times\frac{-2}{5}$
$=\frac{-4}{15}$
View full question & answer→Question 82 Marks
By what number should $(-20)^{-1}$ be divided to obtain $(-10)^{-1}$ ?
AnswerLet the required number be $x.$
$(-20)^{-1}\div \text{x}=(-10)^{-1}$
$\Rightarrow \frac{1}{(-20)}\times \frac{1}{\text{x}}=\frac{1}{(-10)}$
$\Rightarrow \frac{1}{(-20\text{x})}=\frac{1}{(-10)}$
$\therefore \text{x}=\frac{(-10)}{(-20)}=\frac{1}{2}=2^{-1}$
Hence, the required number is $2^{-1}$.
View full question & answer→Question 92 Marks
Express the following in power notation: $(-1)^9$
Answer$(-1)^9$ $=(-1)\times(-1)\times(-1)\times(-1)\times(-1)\times(-1)\\\times(-1)\times(-1)\times(-1)$ $=-1$
View full question & answer→Question 102 Marks
Express the following in power notation: $\Big(\frac{-13}{11}\Big)^2$
Answer$\Big(\frac{-13}{11}\Big)^2=\frac{-13}{11}\times \frac{-13}{11}$ $=\frac{(-13)\times(-13)}{11\times11}$ $=\frac{169}{121}$
View full question & answer→Question 112 Marks
Simplify the following and express a rational number: $\Big(\frac{3}{2}\Big)^4\times \Big(\frac{1}{5}\Big)^2$
Answer$\Big(\frac{3}{2}\Big)^4\times \Big(\frac{1}{5}\Big)^2$ $=\frac{3}{2}\times\frac{3}{2}\times\frac{3}{2}\times\frac{3}{2}\times\frac{1}{5}\times\frac{1}{5}$ $=\frac{81}{400}$
View full question & answer→Question 122 Marks
Express the following in power notation: $\Big(\frac{-1}{2}\Big)^5$
Answer$\Big(\frac{-1}{2}\Big)^5=\frac{-1}{2}\times\frac{-1}{2}\times\frac{-1}{2}\times\frac{-1}{2}\times\frac{-1}{2}$ $=\frac{(-1)\times(-1)\times(-1)\times(-1)\times(-1)}{2\times2\times2\times2\times2}$ $=\frac{-1}{32}$
View full question & answer→Question 132 Marks
By what number should we multiply $(-6)^{-1}$ to abtain a product equal to $9^{-1}$ ?
AnswerLet the required number be $x.$
$(-6)^{-1}\times \text {x}=(9)^{-3}$
$\Rightarrow \frac{1}{-6}\times \text{x}=\frac{1}{9}$
$\therefore \text{x}=\frac{1}{9}\times (-6)=\frac{(-2)}{3}$
Hence, the required number is $\frac{-2}{3}$
View full question & answer→Question 142 Marks
Express the following as a rational number:
$\Big(\frac{5}{7}\Big)^{-1}\times\Big(\frac{7}{4}\Big)^{-1}$
Answer $\Big(\frac{5}{7}\Big)^{-1}\times\Big(\frac{7}{4}\Big)^{-1}$
$=\Big(\frac{7}{5}\Big)^{1}\times\Big(\frac{4}{7}\Big)^{1}$
$=\frac{7}{5}\times\frac{4}{7}=\frac{4}{5}$
View full question & answer→Question 152 Marks
Simplify: $\bigg\{\Big(\frac{-2}{3}\Big)^2\bigg\}^3$
Answer$\bigg\{\Big(\frac{-2}{3}\Big)^2\bigg\}^3$ $=\Big(\frac{-2}{3}\Big)^{2\times3}=\Big(\frac{-2}{3}\Big)^6$ $=\frac{-2}{3}\times\frac{-2}{3}\times\frac{-2}{3}\times\frac{-2}{3}\times\frac{-2}{3}\times\frac{-2}{3}$ $=\frac{64}{729}$
View full question & answer→Question 162 Marks
Simplify: $\Big(\frac{-3}{2}\Big)^{3}\div\Big(\frac{-3}{2}\Big)^{6}$
Answer$\Big(\frac{-3}{2}\Big)^{3}\div\Big(\frac{-3}{2}\Big)^{6}$ $=\Big(\frac{-3}{2}\Big)^{3-6}=\Big(\frac{-3}{2}\Big)^{-3}$ $=\Big(\frac{-2}{3}\Big)^3=\frac{-2}{3}\times\frac{-2}{3}\times\frac{-2}{3}$ $=\frac{-8}{27}$
View full question & answer→Question 172 Marks
Simplify:
$\Big(\frac{-2}{3}\Big)^7\div\Big(\frac{-2}{3}\Big)^4$
Answer $\Big(\frac{-2}{3}\Big)^7\div\Big(\frac{-2}{3}\Big)^4$
$\Big(\frac{-2}{3}\Big)^{7-4}=\Big(\frac{-2}{3}\Big)^3$
$=\frac{-2}{3}\times\frac{-2}{3}\times\frac{-2}{3}$
$=\frac{-8}{27}$
View full question & answer→Question 182 Marks
Find the vallue of n when:
$6^{2 n+1} \div 36=6^3$
Answer$6^{2 n+1} \div 36=6^3$
$\Rightarrow 6^{2 n+1} \div 6^2=6^3$
$\Rightarrow 6^{2 n+1-2}=6^3$
$\Rightarrow 6^{2 n-1}=6^3$
Comparing we get
$2 \mathrm{n}-1=3$
$\Rightarrow 2 \mathrm{n}=3+1=4$
$\Rightarrow \mathrm{n}=\frac{4}{2}=2$
$\therefore \mathrm{n}=2$
View full question & answer→Question 192 Marks
Simplify and express each as a rational number:
$\Big(\frac{4}{9}\Big)^6\times\Big(\frac{4}{9}\Big)^{-4}$
Answer $\Big(\frac{4}{9}\Big)^6\times\Big(\frac{4}{9}\Big)^{-4}$
$=\Big(\frac{4}{9}\Big)^{6-4}=\Big(\frac{4}{9}\Big)^2$
$=\frac{4}{9}\times\frac{4}{9}=\frac{16}{81}$
View full question & answer→Question 202 Marks
By what number should $3^{-3}$ be multiplied to obtain $4?$
AnswerProduct of two numbers $=4$
One number $=3^{-3}=\left(\frac{1}{3}\right)^3$
$=\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}$
$=\frac{1}{27}$
$\therefore$ Second number $=4 \div \frac{1}{27}$
$=4 \times \frac{27}{1}$
$=108$
View full question & answer→Question 212 Marks
Express the following as a rational number: $\Big(\frac{2}{3}\Big)^5$
Answer$\Big(\frac{2}{3}\Big)^5=\frac{2}{3}\times\frac{2}{3}\times\frac{2}{3}\times\frac{2}{3}\times\frac{2}{3}$ $=\frac{2\times2\times2\times2\times2}{3\times3\times3\times3\times3}=\frac{32}{243}$
View full question & answer→Question 222 Marks
Find the vallue of $n$ when:
$5^{2 n} \times 5^3=5^9$
Answer$5^{2 n} \times 5^3=5^9$
$\Rightarrow 5^{2 n+3}=5^9$
Comparing we get,
$2 n+3=9$
$\Rightarrow 2 n=9-3$
$\Rightarrow 2 n=6$
$\Rightarrow n=\frac{6}{2}=3$
$\therefore n=3$
View full question & answer→Question 232 Marks
Express the following as a rational number: $(-3)^{-1}\times\Big(\frac{1}{3}\Big)^{-1}$
Answer$(-3)^{-1}\times\Big(\frac{1}{3}\Big)^{-1}$ $=\Big(\frac{1}{-3}\Big)^{1}\times(3)^{1}$ $=\frac{-1}{3}\times3=-1$
View full question & answer→Question 242 Marks
Express the following as a rational number: $\Big(\frac{-3}{4}\Big)^{-3}$
Answer$\Big(\frac{-3}{4}\Big)^{-3}=\Big(\frac{-4}{3}\Big)^3$ $=\Big(\frac{-4}{3}\Big)\times\Big(\frac{-4}{3}\Big)\times\Big(\frac{-4}{3}\Big)$ $=\frac{(-4)\times(-4)\times(-4)}{3\times3\times3}$ $=\frac{-64}{27}$
View full question & answer→Question 252 Marks
By what number should $(-5)^{-1}$ be multiplied so that the product is $(8)^{-1}$ ?
AnswerProduct of two numbers $=(8)^{-1}=\frac{1}{8}$
One number $=(-5)^{-1}=\frac{-1}{5}$
$\therefore$ Second number $=\frac{1}{8} \div\left(\frac{-1}{5}\right)$
$=\frac{1}{8} \times \frac{-5}{1}$
$=\frac{-5}{8}$
View full question & answer→Question 262 Marks
Simplify: $\Bigg[\bigg\{\Big(\frac{-1}{4}\Big)^2\bigg\}^{-2}\Bigg]^{-1}$
Answer$\Bigg[\bigg\{\Big(\frac{-1}{4}\Big)^2\bigg\}^{-2}\Bigg]^{-1}=\Bigg[\Big(\frac{-1}{4}\Big)^{2\times(-2)}\Bigg]^{-1}$ $=\bigg[\Big(\frac{-1}{4}\Big)^{-4}\bigg]^{-1}$ $=\Big(\frac{-1}{4}\Big)^{-4\times(-1)}=\Big(\frac{-1}{4}\Big)^{4}$ $=\frac{-1}{4}\times\frac{-1}{4}\times\frac{-1}{4}\times\frac{-1}{4}$ $=\frac{1}{256}$
View full question & answer→Question 272 Marks
Express the following as a rational number: $(-2)^{-5}$
Answer$(-2)^{-5}=\Big(\frac{1}{-2}\Big)^{5}$ $=\Big(\frac{1\times(-1)}{-2\times(-1)}\Big)^5=\Big(\frac{-1}{2}\Big)^5$ $=\Big(\frac{(-1)\times(-1)\times(-1)\times(-1)\times(-1)}{2\times2\times2\times2\times2}\Big)$ $=\frac{-1}{32}$
View full question & answer→Question 282 Marks
Simplify and express each as a rational number: $\Big(\frac{-7}{8}\Big)^{-3}\times \Big(\frac{-7}{8}\Big)^2$
Answer$\Big(\frac{-7}{8}\Big)^{-3}\times \Big(\frac{-7}{8}\Big)^2=\Big(\frac{-7}{8}\Big)^{-3+2}$ $=\Big(\frac{-7}{8}\Big)^{-1}=\Big(\frac{8}{-7}\Big)^1=\frac{8}{-7}$ $=\frac{8\times(-1)}{-7\times(-1)}$ $=\frac{-8}{7}$
View full question & answer→Question 292 Marks
Express the following in power notation: $\Big(\frac{-8}{5}\Big)^3$
Answer$\Big(\frac{-8}{5}\Big)^3=\frac{-8}{5}\times\frac{-8}{5}\times\frac{-8}{5}$ $=\frac{(-8)\times(-8)\times(-8)}{5\times 5\times 5}$ $=\frac{-512}{125}$
View full question & answer→Question 302 Marks
Find the vallue of $n$ when:
$8 \times 2^{n+2}=32$
Answer$8 \times 2^{n+2}=32$
$2^{n+2}=\frac{32}{8}$
$4=2^2$
Comparing we get,
$\mathrm{n}+2=2$
$\Rightarrow \mathrm{n}=2-2=0$
$\therefore \mathrm{n}=0$
View full question & answer→Question 312 Marks
Express the following as a rational number:
$\Bigg\{\Big(\frac{4}{3}\Big)^{-1}-\Big(\frac{1}{4}\Big)^{-1}\Bigg\}^{-1}$
Answer $\Bigg\{\Big(\frac{4}{3}\Big)^{-1}-\Big(\frac{1}{4}\Big)^{-1}\Bigg\}^{-1}$
$=\Bigg[\Big(\frac{3}{4}\Big)^{1}-\Big(\frac{4}{1}\Big)\Bigg]^{-1}=\Big[\frac{3}{4}-\frac{4}{1}\Big]^{-1}$
$=\Big[\frac{3-16}{4}\Big]^{-1}=\Big[\frac{-13}{4}\Big]^{-1}$
$=\frac{-4}{13}$
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