[2 Mark Question Answer]

Question 12 Marks

Solve for $x:p^3 \times p^{-2} = p^x$

Answer

$p^3 \times p^{-2}=p^x$
$\Rightarrow p^{3+(-2)}=p^x \quad \ldots \ldots\left(\text { Using } a^m x a^n=a^{m-n}\right)$
$\Rightarrow p^1=p^x$
$\Rightarrow x=1 .$

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Question 22 Marks

Solve for $x:1 = p^x$

Answer

$1 = p^x$
$\Rightarrow p^0 = p^x ......($Using $a^0= 1)$
$\Rightarrow x = 0.$

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Question 32 Marks

Simplify the following$:\frac{5^{ n +2}-6.5^{ n +1}}{13.5^{ n }-2.5^{ n +1}}$

Answer

$\frac{5^{ n +2}-6.5^{ n +1}}{13.5^{ n }-2.5^{ n +1}}$
$ =\frac{5^{ n }\left(5^2-6 \times 5\right)}{5^{ n }(13-2 \times 5)} $
$ =\frac{25-30}{13-10} $
$=\frac{-5}{3} .$

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Question 42 Marks

Evaluate the following$:\left(1-\frac{15}{64}\right)^{-\frac{1}{2}}$

Answer

$\left(1-\frac{15}{64}\right)^{-\frac{1}{2}}$
$ =\left(\frac{64-15}{64}\right)^{-\frac{1}{2}} $
$=\left(\frac{49}{64}\right)^{-\frac{1}{2}}$
$ =\left(\frac{64}{49}\right)^{\frac{1}{2}} $
$=\frac{8}{7} .$

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Question 52 Marks

Simplify the following and express with positive index$:\left[\left( p ^{-3}\right)^{\frac{2}{3}}\right]^{\frac{1}{2}}$

Answer

${\left[\left( p ^{-3}\right)^{\frac{2}{3}}\right]^{\frac{1}{2}}} $
$ = p ^{-3 \times \frac{2}{3} \times \frac{1}{2}} \ldots . .\left(\text { Using }\left(a^m\right)^n=a^{m n}\right)$
$ =p^{-1} $
$=\frac{1}{p} .$

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Question 62 Marks

Write each of the following in the simplest form$:a^2 \times a^3 \div a^4$

Answer

$a^2 \times a^3 \div a^4 $
$ =a^{2+3-4} \ldots\left(\text { Using } a^m \times a^n=a^{m+n} \text { and } a^m \div a^n=a^{m-n}\right)$
$=a^1 $
$=a .$

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Question 72 Marks

Evaluate the following$: \sqrt[6]{25^3}$

Answer

$ \sqrt[6]{25^3}$
$= \sqrt[6]{\left(5^2\right)^3}$
$= \sqrt[6]{5^6}$
$= 5^{6 \times \frac{1}{6}}$
$= 5 .$

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Question 82 Marks