3 Mark Question

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

$\frac{16\times(2)^{\text{n+1}}-4\times2^{\text{n}}}{16\times(2)^{\text{n}+2}-2\times(2)^{\text{n}+2}}$
$=\frac{2^4\times2^{(\text{n}+1)}-2^2\times2^{\text{n}}}{2^4\times2^{(\text{n}+2)}-2^{\text{n}+1}\times2^2}$
$=\frac{2^2\times2^{(\text{n}+3-2\text{n})}}{2^2\times2^{(\text{n}+4-2\text{n}+1)}}$
$=\frac{2^{\text{n}}\times2^3-2^{\text{n}}}{2^{\text{n}}(2^{4}-1)}$
$=\frac{2^{\text{n}}(2^3-1)}{2^{\text{n}}(2^4-1)}$
$=\frac{8-1}{16-2}$
$=\frac{7}{14}$
$=\frac{1}{2}$

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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

$\frac{9^8\times(\text{x}^2)^5}{(27)^4\times(\text{x}^3)^2}$
$=\frac{(3^2)^8\times(\text{x}^2)^5}{(3^3)^4\times(\text{x}^3)^2}$
$=\frac{3^{16}\times\text{x}^{10}}{3^{12}\times\text{x}^6}$
$=3^{16-12}\times\text{x}^{10-6}$
$=3^4\times\text{x}^4$
$=(3\text{x})^{4}$

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Question 33 Marks

Simplify and write the following in exponential form:
$\frac{3^2\times7^8\times13^6}{(21)^2\times(91)^3}$

Answer

$\frac{3^2\times7^8\times13^6}{(21)^2\times(91)^3}$
$=\frac{3^2\times7^27^6\times(13)^6}{(21)^2\times(13)^3\times(7)^3}$
$=\frac{(21)^2\times7^6\times(13)^6}{(21)^2\times(13)^3\times(7)^3}$
$=\frac{7^6\times(13)^6}{(13)^3\times(7)^3}$
$=\frac{91^6}{91^3}$
$=(91)^{6-3}$
$=(91)^3$

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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

$\left(3^5\right)^{11} \times\left(3^{15}\right)^4-\left(3^5\right)^{18} \times\left(3^5\right)^5$
$=3^{55} \times 3^{60}-3^{90} \times 3^{25}$
$=3^{(55+60)}-3^{(90+25)}$
$=3^{(115)}-3^{(115)}$
$=0$

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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

$\frac{10\times(5)^{\text{n}+1}+25\times5^{\text{n}}}{3\times(5)^{\text{n}+2}+10\times(5)^{\text{n}+1}}$
$=\frac{10\times5^{(\text{n}+1)}+5\times5^{(\text{n}+1)}}{3\times5^{(\text{n}+2)}+(2\times5)\times5^{(\text{n}+1)}}$
$=\frac{10\times5^{(\text{n}+1)}+5\times5^{(\text{n}+1)}}{3\times5^{(\text{n}+2)}+(2\times5)\times5^{(\text{n}+1)}}$
$=\frac{5^{(\text{n}+1)}(10+5)}{5^{(\text{n}+1)}(10+15)}$
$=\frac{15}{25}$
$=\frac{3}{5}$

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Question 63 Marks

Simplify:
$\frac{(16)^7\times(25)^5\times(81)^3}{(15)^7\times(24)^5\times(80)^3}$

Answer

$\frac{(16)^7\times(25)^5\times(81)^3}{(15)^7\times(24)^5\times(80)^3}$
$=\frac{(16)^7\times(5^2)^5\times(3^4)^3}{(3\times5)^7\times(3\times8)^5\times(16\times5)^3}$
$=\frac{(16)^7\times(5^2)^5\times(3^4)^3}{3^7\times5^7\times3^5\times8^5\times16^3\times5^3}$
$=\frac{(16)^7\times(5^2)^5\times(3^4)^3}{3^{12}\times5^{10}\times8^5\times16^3}$
$=\frac{(16)^7}{8^5\times16^{3}}$
$=\frac{(16)^4}{8^5}$
$=\frac{(2\times8)^4}{8^5}$
$=\frac{2^4\times8^4}{8^5}$
$=\frac{2^4}{8}$
$=\frac{16}{8}$
$=2$

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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

$=\Big(\frac{2}{3}\Big)^{10}\times\Big(\frac{3}{2}\Big)^{10}=\Big(\frac{2}{3}\Big)^{2\text{n}-2}$
$=\Big(\frac{2}{3}\Big)^{10}\times\Big(\frac{3}{2}\Big)^{10}=\Big(\frac{2}{3}\Big)^{2\text{n}-2}$
$=\frac{2^{10}}{3^{10}}\times\frac{3^{10}}{2^{10}}=\Big(\frac{2}{3}\Big)^{2\text{n}-2}$
$=1=\frac{2^{(2\text{n-2})}}{3^{(2\text{n}-2)}}$
$=3(2\text{n}-2)=2(2\text{n-2})$
$=6\text{n}-6=4\text{n}-4$
$=6\text{n}-4\text{n}=6-4$
$=2\text{n}=2$
$=\text{n}=1$

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