[2 Mark Question Answer]

Question 12 Marks

Solve for $x : (13)^{\sqrt x} = 4^4 - 3^4 - 6$

Answer

$(13)^{\sqrt x} = 4^4 - 3^4 - 6$
$\Rightarrow (13)^{\sqrt x}= 256 - 81 - 6$
$\Rightarrow (13)^{\sqrt x}= 169$
$\Rightarrow (13)^{\sqrt x}= 13^2$
$\Rightarrow \sqrt x = 2$
$\Rightarrow x = 4$

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Question 22 Marks

If $3^{x + 1}= 9^{x - 3},$ find the value of $2^{1 + x}.$

Answer

$3^{x+1}=9^{x-3} $
$\Rightarrow 3^x \times 3=\left(3^2\right)^{x-3} $
$\Rightarrow 3^x \times 3=\frac{3^{2 x}}{3^6} $
$\Rightarrow 3^6 \times 3=\frac{3^{2 x}}{3^x} $
$\Rightarrow 3^7=3^x $
$\Rightarrow x=7 $
$\Rightarrow 2^{1+x}=2^{1+7}$
$=2^8$
$=256$

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Question 32 Marks

Solve : $3^{x-1}\times 5^{2y-3}= 225.$

Answer

$ 3^{x-1} \times 5^{2 y-3}=225$
$ \Rightarrow 3^{x-1} \times 5^{2 y-3}=3^2 \times 5^2$
$ \Rightarrow x-1=2$ and $2 y-3=2$
$ \Rightarrow x=3$ and $2 y=5$
$ \Rightarrow x=3$ and $y=\frac{5}{2}$
$ \Rightarrow x=3$ and $y=2 \frac{1}{2}$

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Question 42 Marks

Evaluate $: \left(\frac{x^q}{x^r}\right)^{\frac{1}{q r}} \times\left(\frac{x^r}{x^p}\right)^{\frac{1}{r p}} \times\left(\frac{x^p}{x^q}\right)^{\frac{1}{p q}}$

Answer

$ \left(\frac{x^q}{x^r}\right)^{\frac{1}{q r}} \times\left(\frac{x^r}{x^p}\right)^{\frac{1}{r p}} \times\left(\frac{x^p}{x^q}\right)^{\frac{1}{p q}}$
$ =\frac{x^{q \times \frac{1}{q r}}}{x^{r \times \frac{1}{q r}}} \times \frac{x^{r \times \frac{1}{r p}}}{x^{p \times \frac{1}{r p}}} \times \frac{x^{p \times \frac{1}{p q}}}{x^{q \times \frac{1}{p q}}}$
$ =\frac{x^{\frac{1}{r}}}{x^{\frac{1}{q}}} \times \frac{x^{\frac{1}{p}}}{x^{\frac{1}{r}}} \times \frac{x^{\frac{1}{q}}}{x^{\frac{1}{p}}}$
$ =1$

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Question 52 Marks

Solve :$4^{x - 2}- 2^{x + 1}= 0$

Answer

$4^{x - 2}- 2^{x + 1}= 0$
$\Rightarrow 4^{x - 2}= 2^{x + 1}$
$\Rightarrow (2^2)^{x - 2}= 2^{x + 1}$
$\Rightarrow (2^2)^{x - 4}= 2^{x + 1}$
We know that if bases are equal, the powers are equal
$\Rightarrow 2x - 4 = x + 1$
$\Rightarrow 2x - x = 4 + 1$
$\Rightarrow x = 5.$

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Question 62 Marks

Solve for $x : 9^{x+2}= 720 + 9^x$

Answer

$9^{x+2}= 720 + 9^x$
$\Rightarrow 9^{x+2}- 9^x= 720$
$\Rightarrow 9^x(9^2- 1) = 720$
$\Rightarrow 9^x(81 - 1) = 720$
$\Rightarrow 9^x(80) = 720$
$\Rightarrow 9^x= 9$
$\Rightarrow 9^x= 9^1$
$\Rightarrow x = 1$

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Question 72 Marks

Solve for $x : 3(2^x+ 1) - 2^{x + 2}+ 5 = 0$

Answer

$3(2^x+ 1) - 2^{x + 2}+ 5 = 0$
$\Rightarrow 3 \times 2^x + 3 - 2^x \times 2^2 + 5 = 0$
$\Rightarrow 2^x ( 3 - 4 ) + 8 = 0$
$\Rightarrow - 2^x= - 8$
$\Rightarrow 2^x= 2^3$
$\Rightarrow x = 3$

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Question 82 Marks

Solve for $x :2^{3x + 3}= 2^{3x + 1}+ 48$

Answer

$2^{3x + 3}= 2^{3x + 1}+ 48$
$\Rightarrow 8 \times 2^{3x} = 2^{3x} \times 2 + 48$
$\Rightarrow 2^{3x} ( 8 - 2 ) = 48$
$\Rightarrow 2^{3x} \times 6 = 48$
$\Rightarrow 2^{3x} = 8$
$\Rightarrow 2^{3x} = 2^3$
$\Rightarrow 3x = 3$
$\Rightarrow x = 1$

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Question 92 Marks

Solve for ${x}:(81)^{\frac{3}{4}}-\left(\frac{1}{32}\right)^{-\frac{2}{5}}+x\left(\frac{1}{2}\right)^{-1} .2^0=27$

Answer

$ (81)^{\frac{3}{4}}-\left(\frac{1}{32}\right)^{-\frac{2}{5}}+x\left(\frac{1}{2}\right)^{-1} .2^0=27$
$ \Rightarrow 3^{4 \times \frac{3}{4}}-\left(2^{-} 5\right)^{-\frac{2}{5}}+x(2)=27$
$ \Rightarrow 3^3-2^2+2 x=27$
$ \Rightarrow 2 x+27-4=27$
$ \Rightarrow 2 x=4$
$ \Rightarrow x=2$

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Question 102 Marks

Solve for $x : (a^{3x + 5})^2. (a^x)^4= a^{8x + 12}$

Answer

$(a^{3x + 5})^2. (a^x)^4= a^{8x + 12}$
$\Rightarrow a^{6x + 10 + 4x =}a^{8x + 12}$
If bases are the same, powers are also same
$\Rightarrow 10x + 10 = 8x + 12$
$\Rightarrow 2x = 2$
$\Rightarrow x = 1$

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Question 112 Marks

Solve for $x : 3^{4x + 1}= (27)^{x + 1}$

Answer

$3^{4x + 1}= (27)^{x + 1}$
$\Rightarrow 3^{4x + 1} = (3^3)^{x + 1}$
$\Rightarrow 3^{4x + 1} =( 3^3)^{x + 1}$
We know that if bases are equal, the powers are equal
$\Rightarrow 4x + 1 = 3x + 3$
$\Rightarrow 4x - 3x = 3 - 1$
$\Rightarrow x = 2.$

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Question 122 Marks

Solve for $x : 2^{2x+1}= 8$

Answer

$2^{2 x+1}=8$
$\Rightarrow 2^{2 x+1}=2^3$
We know that if bases are equal, the powers are equal
$ \Rightarrow 2 x+1=3 $
$ \Rightarrow 2 x=3-1 $
$ \Rightarrow 2 x=2 $
$ \Rightarrow x=\frac{2}{2} $
$ \Rightarrow x=1$

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Question 132 Marks

Simplify the following and express with positive index $:\left(\frac{3^{-4}}{2^{-8}}\right)^{\frac{1}{4}}$

Answer

$ \left(\frac{3^{-4}}{2^{-8}}\right)^{\frac{1}{4}}$
$ =\left(\frac{2^8}{3^4}\right)^{\frac{1}{4}}$
$ =\frac{\left(2^8\right)^{\frac{1}{4}}}{\left(3^4\right)^{\frac{1}{4}}}$
$ =\frac{2^{8 \times \frac{1}{4}}}{3^{4 \times \frac{1}{4}}}$
$ =\frac{2^2}{3}$
$ =\frac{4}{3}$

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Question 142 Marks

Simplify $ :\left(3 x^2\right)^{-3} \times\left(x^9\right)^{\frac{2}{3}}$

Answer

$ \left(3 x^2\right)^{-3} \times\left(x^9\right)^{\frac{2}{3}}$
$ =\frac{1}{\left(3 x^2\right)^3} \times x^{9 \times \frac{2}{3}}$
$ =\frac{1}{3^3 x^{2 \times 3}} \times x^6$
$ =\frac{1}{27 x^6} \times x^6$
$ =\frac{1}{27}$

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Question 152 Marks

Simplify $:\frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^n-5^n \times 2^2}$

Answer

$ \frac{5^{n+3}-6 \times 5^{n+1}}{9 \times 5^n-5^n \times 2^2}$
$ =\frac{5^{n+1} \times 5^2-6 \times 5^{n+1}}{9 \times 5^n-5^n \times 2^2}$
$ =\frac{5^{n+1} \times\left(5^2-6\right)}{5^n \times(9-4)}$
$ =\frac{5^1 \times 19}{5}$
$ =19$

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Question 162 Marks

Simplify $:(a+b)^{-1} \cdot\left(a^{-1}+b^{-1}\right)$

Answer

$ (a+b)^{-1} \cdot\left(a^{-1}+b^{-1}\right)$
$ =\frac{1}{a+b} \times\left(\frac{1}{a}+\frac{1}{b}\right)$
$ =\frac{1}{a+b} \times\left(\frac{b+a}{a b}\right)$
$ =\frac{1}{a+b} \times \frac{a+b}{a b}$
$ =\frac{1}{a b}$

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