[3 marks sum]

Question 13 Marks

If $m^2 = -2$ and $n = 2$; find the values of: $2n^3 – 3m$

Answer

$2n^3 – 3m$
$m = -2, n = 2$
$= 2(2)^3 - 3(-2)$
$= 2 \times (2 \times 2 \times 2) - 3 \times (- 2)$
$= 16 - 3 \times (-2)$
$= 16 + 6 = 22$

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Question 23 Marks

If $m^2 = -2$ and $n = 2$; find the values of: $6m^{-3} + 4n^2$

Answer

$\begin{aligned} & 6 m^{-3}+4 n^2 \\ & m=-2, n=2 \\ & =6(-2)^{-3}+4(2)^2 \\ & =6 \times \frac{1}{-2} \times \frac{1}{-2} \times \frac{1}{-2}+4 \times 2 \times 2 \\ & =\frac{-3}{4}+16 \\ & =\frac{-3+16 \times 4}{4}=\frac{-3+64}{4}=\frac{61}{4}=15 \frac{1}{4}\end{aligned}$

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Question 33 Marks

Evaluate: $2^5 \times 15^0+(-3)^3-\left(\frac{2}{7}\right)^{-2}$

Answer

$\begin{aligned} & 2^5 \times 15^0+(-3)^3-\left(\frac{2}{7}\right)^{-2} \\ & =2 \times 2 \times 2 \times 2 \times 2 \times 1+(-3) \times(-3) \times(-3)-\left(\frac{7}{2}\right) \times\left(\frac{7}{2}\right) \\ & =32 \times 1-27-\frac{49}{4}\left(\because a^0=1\right) \\ & =\frac{32 \times 4}{1 \times 4}-\frac{27 \times 4}{1 \times 4}-\frac{49}{4 \times 1}(\because \text { LCM }=4) \\ & =(128-108-49) / 4=-\frac{29}{4}=-7 \frac{1}{4}\end{aligned}$

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Question 43 Marks

Evaluate: $6^{-2} \div\left(4^{-2} \times 3^{-2}\right)$

Answer

$\begin{aligned} & 6^{-2} \div\left(4^{-2} \times 3^{-2}\right) \\ & =\left(\frac{1}{6}\right)^2 \div\left(\frac{1}{4}\right)^2 \times\left(\frac{1}{3}\right)^2 \\ & =\frac{1}{36} \div \frac{1}{16} \times \frac{1}{9} \\ & =\frac{1}{36} \div \frac{1}{144}\end{aligned}$

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Question 53 Marks

Simplify and express the Solution in the positive exponent form:
$\frac{\left(2^3\right)^5 \times 5^4}{4^3 \times 5^2}$

Answer

$\begin{aligned} & \frac{\left(2^3\right)^5 \times 5^4}{4^3 \times 5^2} \\ & =\frac{2^{3 \times 5} \times 5^4}{2^3 \times 2^2 \times 5^2} \\ & =\frac{2^{15} \times 5^4}{2^6 \times 5^2} \\ & =2^{15-6} \times 5^{4-2} \\ & =2^9 \times 5^2\end{aligned}$

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Question 63 Marks

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

Answer

$\begin{aligned} & \frac{(-3)^3 \times 2^6}{6 \times 2^3} \\ & =\frac{(-3)^3 \times 2^6}{2 \times 3 \times 2^3} \\ & =\frac{(-3)^3 \times 2^6}{3 \times 2^{3+1}} \\ & =-(3)^{3-1} 2^{6-4} \\ & =-(3)^2 2^2 \\ & =-3^2 2^2\end{aligned}$

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Question 73 Marks

Simplify, giving Solution with positive index
$\frac{\left(5 x^7\right)^3 \cdot\left(10 x^2\right)^2}{\left(2 x^6\right)^7}=\frac{5^3 x^{7 \times 3} \cdot 10^2 \cdot x^{2 \times 2}}{2^7 \cdot x^{6 \times 7}}$

Answer

$\begin{aligned} & \frac{\left(5 \mathrm{x}^7\right)^3 \cdot\left(10 \mathrm{x}^2\right)^2}{\left(2 \mathrm{x}^6\right)^7}=\frac{5^3 \mathrm{x}^{7 \times 3} \cdot 10^2 \cdot \mathrm{x}^{2 \times 2}}{2^7 \cdot \mathrm{x}^{6 \times 7}} \\ & =5^3 \cdot 10^2 \cdot 2^{-7} \mathrm{x}^{21+4-42} \\ & =\frac{5^3 \times 10^2}{2^7 \mathrm{x}^{17}}=\frac{5 \times 5 \times 5 \times 2 \times 5 \times 2 \times 5}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \cdot \mathrm{x}^{17}} \\ & =\frac{5^5}{2^5 \mathrm{x}^{17}}=\frac{3125}{32 \mathrm{x}^{17}}\end{aligned}$

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Question 83 Marks

Simplify, giving a Solution with a positive index.
$\left(\frac{1}{2 x}\right)^3 \times(6 x)^2$

Answer

$\begin{aligned} & \left(\frac{1}{2 \mathrm{x}}\right)^3 \times(6 \mathrm{x})^2 \\ & =\frac{1^3}{(2 x)^3} \times 6^2 \times x^2 \\ & =\frac{1}{2^3 \times x^3} \times 36 \times x^2 \\ & =\frac{36 x^2}{8 x^3} \\ & =\frac{9}{2} x^{2-3} \\ & =\frac{9}{2} x^{-1}=\frac{9}{2 x}\end{aligned}$

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Question 93 Marks

Simplify, giving Solution with positive index
$(2a^3)^4 (4a^2)^2$

Answer

$(2a^3)^4 (4a^2)^2$
$= (2a^3)^4 (2^2a^2)^2$
$= 2^4 a^{3\times 4} . 2^{2\times 2} a^{2\times 2}$
$= 2^4a^{12} . 2^4a^4$
$= 2^{4+4} a^{12+4}$
$= 2^8a^{16}$
$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times a^{16}$
$= 256 a^{16}$

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Question 103 Marks

Simplify, giving Solution with positive index
$- (3ab)^2 (-5a^2bc^4)^2$

Answer

$\begin{aligned} & -(3 a b)^2\left(-5 a^2 b c^4\right)^2 \\ & =-\left(3^2 a^2 b^2\right) \times(-1)^2 \times 5^2 a^{2 \times 2} b^2 c^{4 \times 2} \\ & =-\left(3^2 a^2 b^2\right)\left(5^2 a^4 b^2 c^8\right) \\ & =-3^2 5^2 a^{2+4} b^{2+2} c^8 \\ & =-225 a^6 b^4 c^8\end{aligned}$

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Question 113 Marks

Simplify, giving Solution with positive index
$(- 4x) (-5x^2)$

Answer

$\begin{aligned} & (-4 x)\left(-5 x^2\right) \\ & =(-1 \times 4 \times x) \cdot\left(-1 \times 5 \times x^2\right)^1 \\ & =-1 \times 4 \times x \cdot-1 \times 5 \times x^2 \\ & =-1 \times-1 \times 4 \times 5 \times x^{1+2} \\ & =-1^{1+1} \cdot 4^1 \cdot 5^1 x^3=20 x^3\end{aligned}$

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Question 123 Marks

If $27 \times 32 = 3^x \times 2^y;$ find the values of $x$ and $y$.
$27 \times 32 = 3^x \times 2^y$
$27 = 3^x$

$3$	$27$
$3$	$9$
$3$	$3$
	$1$

Answer

$27 = 3 \times 3 \times 3 = 3^3 = 3^x$
$\therefore x = 3^x$
Also, $32 = 2^y$

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

$32 = 2 \times 2 \times 2 \times 2 \times 2$
$= 2^5 = 2^y$
$\therefore y = 5$

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Question 133 Marks

Express: $729$ as a power of $3.$

Answer

$729$ as a power of $3.$

$3$	$729$
$3$	$243$
$3$	$81$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

$= 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6$

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Question 143 Marks

Express: $343$ as a power of $7.$

Answer

$343$ as a power of $7.$
$= 7 \times 7 \times 7 = 7^3$

$7$	$343$
$7$	$49$
$7$	$7$
	$1$

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Question 153 Marks

Express the following in exponential form: $1176$

Answer

$1176$

$2$	$1176$
$2$	$588$
$2$	$294$
$3$	$147$
$7$	$49$
$7$	$7$
	$1$

$= 2 \times 2 \times 2 \times 3 \times 7 \times 7$
$= 2^3 \times 3 \times 7^2 $

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Question 163 Marks

Express the following in exponential form: $1350$

Answer

$1350$

$2$	$1350$
$3$	$675$
$3$	$225$
$3$	$75$
$5$	$25$
$5$	$5$
	$1$

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

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Question 173 Marks

Express the following in exponential form: $1458$

Answer

1458

$2$	$1458$
$3$	$729$
$3$	$243$
$3$	$81$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

$= 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2 \times 3^6 $

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Question 183 Marks

Evaluate: $\left(\frac{-3}{-5}\right)^3$

Answer

$\begin{aligned} & \left(\frac{-3}{-5}\right)^3 \\ & =\left(\frac{-3}{-5}\right) \times\left(\frac{-3}{-5}\right) \times\left(\frac{-3}{-5}\right) \\ & =\frac{(-3) \times(-3) \times(-3)}{(-5) \times(-5) \times(-5)} \\ & =\frac{27}{125}\end{aligned}$

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Question 193 Marks

Evaluate: $\left(-\frac{5}{6}\right)^5$

Answer

$\begin{aligned} & \left(-\frac{5}{6}\right)^5 \\ & =\left(\frac{-5}{6}\right) \times\left(\frac{-5}{6}\right) \times\left(\frac{-5}{6}\right) \times\left(\frac{-5}{6}\right) \times\left(\frac{-5}{6}\right) \\ & =\frac{(-5) \times(-5) \times(-5) \times(-5) \times(-5)}{6 \times 6 \times 6 \times 6 \times 6} \\ & =-\frac{3125}{776}\end{aligned}$

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MATHS [3 marks sum]

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