[4 marks sum]

Question 14 Marks

Evaluate: $\left(2^2\right)^0+2^{-4} \div 2^{-6}+\left(\frac{1}{2}\right)^{-3}$

Answer

$\begin{aligned} & \left(2^2\right)^0+2^{-4} \div 2^{-6}+\left(\frac{1}{2}\right)^{-3} \\ & =(4)^0+\left(\frac{1}{2}\right)^4 \div\left(\frac{1}{2}\right)^6+\left(\frac{2}{1}\right)^3\left(\because a^0=1\right) \\ & =1+\left(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\right) \div\left(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\right)+\left(\frac{2}{1} \times \frac{2}{1} \times \frac{2}{1}\right) \\ & =1+\left(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\right)+8 \\ & =1+4+8=13\end{aligned}$

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Question 24 Marks

Simplify and express the Solution in the positive exponent form:
$\frac{36 \times(-6)^2 \times 3^6}{12^3 \times 3^5}$

Answer

$\frac{36 \times(-6)^2 \times 3^6}{12^3 \times 3^5}$
$=\frac{6 \times 6 \times(-6)^2 \times 3^6}{3^3 \times 4^3 \times 3^5}$
$=\frac{(6)^2(-6)^2 \times 3^{6-3-5}}{4^3}$
$=\frac{(6)^2(-6)^2 3^{-2}}{4^3}$
$=\frac{6^2(-6)^2}{3^2 \times 4^3}$
$=\frac{6 \times 6 \times-6 \times-6}{3 \times 3 \times 4 \times 4 \times 4}$
$=\frac{9}{4}=\left(\frac{3}{2}\right)^2$

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Question 34 Marks

Simplify, giving Solution with positive index
$\frac{\left(7 p^2 q^9 r^5\right)^2(4 p q r)^3}{\left(14 p^6 q^{10} r^4\right)^2}$

Answer

$\begin{aligned} & \frac{\left(7 p^2 q^9 r^5\right)^2(4 p q r)^3}{\left(14 p^6 q^{10} r^4\right)^2} \\ & =\frac{7^2 p^{2 \times 2} q^{9 \times 2} r^{5 \times 2}\left(4^3 p^3 q^3 r^3\right)}{14^2 p^{6 \times 2} q^{10 \times 2} r^{4 \times 2}} \\ & =\frac{7 \times 7 p^4 q^{18} r^{10} \cdot 4 \times 4 \times 4 p^3 q^3 r^3}{2 \times 7 \times 2 \times 7 \times p^{12} q^{20} r^8} \\ & =p^{4-12+3} q^{18-20+3} r^{10-8+3} 4 \times 4 \\ & =16 p^{-5} q^5 \\ & =\frac{16 q r^5}{p^5}\end{aligned}$

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Question 44 Marks

Simplify, giving Solution with positive index
$\left(\frac{1}{4 a b^2 c}\right)^2 \div\left(\frac{3}{2 a^2 b^2}\right)^4$

Answer

$\begin{aligned} & \left(\frac{1}{4 a b^2 c}\right)^2 \div\left(\frac{3}{2 a^2 b c^2}\right)^4 \\ & =\left(\frac{1}{4 a b^2} c\right)^2 \times\left(\frac{2 a^2 b c^2}{3}\right)^4 \\ & =\frac{1^2}{4^2 a^2 b^{2 \times 2} \cdot c^2} \times \frac{2^4 a^{2 \times 4} \cdot b^4 \cdot c^{2 \times 4}}{3^4} \\ & =\frac{1^2}{3^4} \times a^{8-2} b^{4-4} c^{8-2}\left(\because 2^4=4^2\right) \\ & =\frac{1}{3 \times 3 \times 3 \times 3} a^6 b^0 c^6 \\ & =\frac{1}{81} a^6 c^6\left(\because b^0=1\right)\end{aligned}$

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Question 54 Marks

Express: $1024$ as a power of $2.$

Answer

$1024$ as a power of $2.$

$2$	$1024$
$2$	$512$
$2$	$256$
$2$	$128$
$2$	$64$
$2$	$32$
$2$	$16$
$2$	$8$
$2$	$4$
$2$	$2$
	$1$

$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{10}$

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Question 64 Marks

Express the following in exponential form: $3600$

Answer

$3600$

$2$	$3600$
$2$	$1800$
$2$	$900$
$2$	$450$
$3$	$225$
$3$	$75$
$5$	$25$
$5$	$5$
	$1$

$= 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5$
$= 2^4 \times 3^2 \times 5^2 $

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Question 74 Marks

Express the following in exponential form: $1250$

Answer

$1250$

$2$	$1250$
$5$	$625$
$5$	$125$
$5$	$25$
$5$	$5$
	$1$

$= 2 \times 5 \times 5 \times 5 \times 5 = 2 \times 5^4 $

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Question 84 Marks

Express the following in exponential form: $512$

Answer

$512$

$2$	$512$
$2$	$256$
$2$	$128$
$2$	$64$
$2$	$32$
$2$	$16$
$2$	$8$
$2$	$4$
$2$	$2$
	$1$

$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9 $

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Question 94 Marks

Evaluate: $\left[\left(\frac{5}{6}\right)^2 \times \frac{9}{4}\right] \div\left[\left(-\frac{3^2}{2}\right) \times \frac{125}{216}\right]$

Answer

$\begin{aligned} & {\left[\left(\frac{5}{6}\right)^2 \times \frac{9}{4}\right] \div\left[\left(-\frac{3^2}{2}\right) \times \frac{125}{216}\right]} \\ & =\left[\frac{5 \times 5}{6 \times 6} \times \frac{9}{4}\right] \div\left[\frac{-3 \times-3}{2 \times 2} \times \frac{125}{216}\right] \\ & =\left[\frac{25}{36} \times \frac{9}{4}\right] \div\left[\frac{9}{4} \times \frac{125}{216}\right] \\ & =\left[\frac{25}{4} \times \frac{1}{4}\right] \div\left[\frac{1}{4} \times \frac{125}{24}\right] \\ & =\left[\frac{25}{16}\right] \div\left[\frac{125}{96}\right] \\ & =\frac{25}{16} \times \frac{96}{125} \\ & =\frac{1}{1} \times \frac{6}{5}=1 \frac{1}{5}\end{aligned}$

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Question 104 Marks

Simplify and express the Solution in the positive exponent form:
$-\frac{128}{2187}$

Answer

$-\frac{128}{2187}$

2	128
2	64
2	32
2	16
2	8
2	4
2	2
	1

3	2187
3	729
3	243
3	81
3	27
3	9
3	3
	1

$-\frac{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}{3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3}=-\frac{2^7}{3^7}$

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Question 114 Marks

If $64 \times 625 = 2^a \times 5^b ;$ find : the values of $a$ and $b$.

Answer

the values of $a$ and $b$.
$64 \times 625 = 2^a \times 5^b$
$64 = 2^a$

$2$	$64$
$2$	$32$
$2$	$16$
$2$	$8$
$2$	$4$
$2$	$2$
	$1$

$64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2$
$64 = 2^6$
$\therefore a = 6$
Also$, 625 = 5^b$

$5$	$625$
$5$	$125$
$5$	$25$
$5$	$5$
	$1$

$625 = 5 \times 5 \times 5 \times 5$
$625 = 5^4$
$\therefore b = 4$

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MATHS [4 marks sum]

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