Question 13 Marks
Simplify: $\frac{16\times(2)^{\text{n+1}}-4\times2^{\text{n}}}{16\times(2)^{\text{n}+2}-2\times(2)^{\text{n}+2}}$
Answer$\frac{16\times(2)^{\text{n+1}}-4\times2^{\text{n}}}{16\times(2)^{\text{n}+2}-2\times(2)^{\text{n}+2}}$
$=\frac{2^4\times2^{(\text{n}+1)}-2^2\times2^{\text{n}}}{2^4\times2^{(\text{n}+2)}-2^{\text{n}+1}\times2^2}$
$=\frac{2^2\times2^{(\text{n}+3-2\text{n})}}{2^2\times2^{(\text{n}+4-2\text{n}+1)}}$
$=\frac{2^{\text{n}}\times2^3-2^{\text{n}}}{2^{\text{n}}(2^{4}-1)}$
$=\frac{2^{\text{n}}(2^3-1)}{2^{\text{n}}(2^4-1)}$
$=\frac{8-1}{16-2}$
$=\frac{7}{14}$
$=\frac{1}{2}$
View full question & answer→Question 23 Marks
Simplify and write the following in exponential form: $\frac{9^8\times(\text{x}^2)^5}{(27)^4\times(\text{x}^3)^2}$
Answer$\frac{9^8\times(\text{x}^2)^5}{(27)^4\times(\text{x}^3)^2}$
$=\frac{(3^2)^8\times(\text{x}^2)^5}{(3^3)^4\times(\text{x}^3)^2}$
$=\frac{3^{16}\times\text{x}^{10}}{3^{12}\times\text{x}^6}$
$=3^{16-12}\times\text{x}^{10-6}$
$=3^4\times\text{x}^4$
$=(3\text{x})^{4}$
View full question & answer→Question 33 Marks
Simplify and write the following in exponential form:
$\frac{3^2\times7^8\times13^6}{(21)^2\times(91)^3}$
Answer$\frac{3^2\times7^8\times13^6}{(21)^2\times(91)^3}$
$=\frac{3^2\times7^27^6\times(13)^6}{(21)^2\times(13)^3\times(7)^3}$
$=\frac{(21)^2\times7^6\times(13)^6}{(21)^2\times(13)^3\times(7)^3}$
$=\frac{7^6\times(13)^6}{(13)^3\times(7)^3}$
$=\frac{91^6}{91^3}$
$=(91)^{6-3}$
$=(91)^3$
View full question & answer→Question 43 Marks
Simplify:
$\left(3^5\right)^{11} \times\left(3^{15}\right)^4-\left(3^5\right)^{18} \times\left(3^5\right)^5$
Answer$\left(3^5\right)^{11} \times\left(3^{15}\right)^4-\left(3^5\right)^{18} \times\left(3^5\right)^5$
$=3^{55} \times 3^{60}-3^{90} \times 3^{25}$
$=3^{(55+60)}-3^{(90+25)}$
$=3^{(115)}-3^{(115)}$
$=0$
View full question & answer→Question 53 Marks
Simplify: $\frac{10\times(5)^{\text{n}+1}+25\times5^{\text{n}}}{3\times(5)^{\text{n}+2}+10\times(5)^{\text{n}+1}}$
Answer$\frac{10\times(5)^{\text{n}+1}+25\times5^{\text{n}}}{3\times(5)^{\text{n}+2}+10\times(5)^{\text{n}+1}}$
$=\frac{10\times5^{(\text{n}+1)}+5\times5^{(\text{n}+1)}}{3\times5^{(\text{n}+2)}+(2\times5)\times5^{(\text{n}+1)}}$
$=\frac{10\times5^{(\text{n}+1)}+5\times5^{(\text{n}+1)}}{3\times5^{(\text{n}+2)}+(2\times5)\times5^{(\text{n}+1)}}$
$=\frac{5^{(\text{n}+1)}(10+5)}{5^{(\text{n}+1)}(10+15)}$
$=\frac{15}{25}$
$=\frac{3}{5}$
View full question & answer→Question 63 Marks
Simplify:
$\frac{(16)^7\times(25)^5\times(81)^3}{(15)^7\times(24)^5\times(80)^3}$
Answer $\frac{(16)^7\times(25)^5\times(81)^3}{(15)^7\times(24)^5\times(80)^3}$
$=\frac{(16)^7\times(5^2)^5\times(3^4)^3}{(3\times5)^7\times(3\times8)^5\times(16\times5)^3}$
$=\frac{(16)^7\times(5^2)^5\times(3^4)^3}{3^7\times5^7\times3^5\times8^5\times16^3\times5^3}$
$=\frac{(16)^7\times(5^2)^5\times(3^4)^3}{3^{12}\times5^{10}\times8^5\times16^3}$
$=\frac{(16)^7}{8^5\times16^{3}}$
$=\frac{(16)^4}{8^5}$
$=\frac{(2\times8)^4}{8^5}$
$=\frac{2^4\times8^4}{8^5}$
$=\frac{2^4}{8}$
$=\frac{16}{8}$
$=2$
View full question & answer→Question 73 Marks
Find the values of n in the following:
$\Big(\frac{3}{2}\Big)^4\times\Big(\frac{3}{2}\Big)^5=\Big(\frac{3}{2}\Big)^{2\text{n}+1}$
Answer $=\Big(\frac{2}{3}\Big)^{10}\times\Big(\frac{3}{2}\Big)^{10}=\Big(\frac{2}{3}\Big)^{2\text{n}-2}$
$=\Big(\frac{2}{3}\Big)^{10}\times\Big(\frac{3}{2}\Big)^{10}=\Big(\frac{2}{3}\Big)^{2\text{n}-2}$
$=\frac{2^{10}}{3^{10}}\times\frac{3^{10}}{2^{10}}=\Big(\frac{2}{3}\Big)^{2\text{n}-2}$
$=1=\frac{2^{(2\text{n-2})}}{3^{(2\text{n}-2)}}$
$=3(2\text{n}-2)=2(2\text{n-2})$
$=6\text{n}-6=4\text{n}-4$
$=6\text{n}-4\text{n}=6-4$
$=2\text{n}=2$
$=\text{n}=1$
View full question & answer→Question 83 Marks
Find the number from the following expanded form:
$3 \times 10^4+4 \times 10^2+5 \times 10^0$
View full question & answer→Question 93 Marks
Find the number from the following expanded form:
$9 \times 10^5+5 \times 10^2+3 \times 10^1$
View full question & answer→Question 103 Marks
Find the number from the following expanded form:
$5 \times 10^5+4 \times 10^4+2 \times 10^3+3 \times 10^0$
View full question & answer→Question 113 Marks
Find the number from the following expanded form:
$7 \times 10^4+6 \times 10^3+0 \times 10^2+4 \times 10^1+5 \times 10^0$
View full question & answer→Question 123 Marks
Write the following number in the expanded exponential forms:
927303
Answer$9 \times 10^5+2 \times 10^4+7 \times 10^3+3 \times 10^2+0 \times 10^1+3 \times 10^0$
View full question & answer→Question 133 Marks
Write the following number in the expanded exponential forms:
5004132
Answer$5 \times 10^6+0 \times 10^5+0 \times 10^4+4 \times 10^3+1 \times 10^2+3 \times 10^1+2 \times 10^0$
View full question & answer→Question 143 Marks
Write the following number in the expanded exponential forms:
7805192
Answer$7 \times 10^6+8 \times 10^5+0 \times 10^4+5 \times 10^3+1 \times 10^2+9 \times 10^1+2 \times 10^0$
View full question & answer→Question 153 Marks
Write the following number in the expanded exponential forms:
420719
Answer$4 \times 10^5+2 \times 10^4+0 \times 10^3+7 \times 10^2+1 \times 10^1+9 \times 10^0$
View full question & answer→Question 163 Marks
Write the following number in the expanded exponential forms:
20068
Answer$2 \times 10^4+0 \times 10^3+0 \times 10^2+6 \times 10^1+8 \times 10^0$
View full question & answer→Question 173 Marks
Express the numbers appearing in the following statements in the standard form:
The universe is estimated to be about $12,000,000,000$ years old.
Answer$1.2 \times 10^{10}$ years old
View full question & answer→Question 183 Marks
Express the numbers appearing in the following statements in the standard form:
Diameter of the Sun is $1,400,000,000$ metres.
Answer$1.4 \times 10^9 m$
View full question & answer→Question 193 Marks
Express the numbers appearing in the following statements in the standard form:
Diameter of the Earth is $1,27,56,000$ metres.
Answer$1.2756 \times 10^7 m$
View full question & answer→Question 203 Marks
Express the numbers appearing in the following statements in the standard form:
The distance between the Earth and the Moon is $384,000,000$ metres.
Answer$3.84 \times 10^8 m$
View full question & answer→Question 213 Marks
Find the values of $n$ in the following:
$\left(\frac{2}{3}\right)^{10} \times\left\{\left(\frac{3}{2}\right)^2\right\}^5=\left(\frac{2}{3}\right)^{2 n-2}$
View full question & answer→Question 223 Marks
Find the values of $n$ in the following:
$\left(\frac{3}{2}\right)^4 \times\left(\frac{3}{2}\right)^5=\left(\frac{3}{2}\right)^{2 n+1}$
View full question & answer→Question 233 Marks
Find the values of $n$ in the following:
$7^{2 n+1} \div 49=7^3$
View full question & answer→Question 243 Marks
If $a=2$ and $b=3$, then find the values of the following:
$\left(\frac{a}{b}+\frac{b}{a}\right)^a$
View full question & answer→Question 253 Marks
If $a=2$ and $b=3$, then find the values of the following:
$\left(\frac{b}{a}\right)^b$
View full question & answer→Question 263 Marks
If $a=2$ and $b=3$, then find the values ofthe following:
$(a b)^b$
View full question & answer→Question 273 Marks
Express the following number as a product of powers of their prime factors:
24000
Answer$2^6 \times 3^1 \times 5^3$
View full question & answer→Question 283 Marks
Express the following number as a product of powers of their prime factors:
2800
Answer$2^4 \times 5^2 \times 7$
View full question & answer→Question 293 Marks
Express the following number as a product of powers of their prime factors:
450
Answer$2 \times 3^2 \times 5^2$
View full question & answer→Question 303 Marks
Express the following number as a product of powers of their prime factors:
392
Question 313 Marks
Express the following number as a product of powers of their prime factors:
675
Question 323 Marks
Express the following number as a product of powers of their prime factors:
36