Question 13 Marks
Solve for $x:9^{x+4}= 3^2 x (27)^{x+1}$
Answer$9^{x+4}=3^2 \times(27)^{x+1}$
$ \Rightarrow 9^{x+4}=3^2 \times\left(3^3\right)^{x+1}$
$ \Rightarrow 3^{2(x+4)}=3^2 \times 3^{3 x+3} $
$ \Rightarrow 3^{2 x+8}=3^{2+3 x+3} $
$\Rightarrow 2 x+8=2+3 x+3 $
$ \Rightarrow 2 x+8=3 x+5 $
$ \Rightarrow x=3$
View full question & answer→Question 23 Marks
Solve for $x:\sqrt{\left(\frac{3}{5}\right)^{x+3}}=\frac{27^{-1}}{125^{-1}}$
Answer$\sqrt{\left(\frac{3}{5}\right)^{x+3}}=\frac{27^{-1}}{125^{-1}}$
$\Rightarrow\left(\frac{3}{5}\right)^{(x+3) \times\left(\frac{1}{2}\right)}=\frac{\left(3^3\right)^{-1}}{\left(5^3\right)-1}$
$\Rightarrow\left(\frac{3}{5}\right)^{\frac{x+3}{2}}=\left(\frac{3}{5}\right)^{-3}$
$\Rightarrow \frac{x+3}{2}=-3$
$\Rightarrow x+3=-6$
$\Rightarrow x=-9 .$
View full question & answer→Question 33 Marks
Solve for $x:\sqrt{\left(8^0+\frac{2}{3}\right)}=(0.6)^{2-3 x}$
Answer$ \sqrt{\left(8^0+\frac{2}{3}\right)}=(0.6)^{2-3 x}$
$\Rightarrow \left(1+\frac{2}{3}\right)^{\frac{1}{2}}=\left(\frac{6}{10}\right)^{2-3 x}$
$\Rightarrow \left(\frac{5}{3}\right)^{\frac{1}{2}}=\left(\frac{3}{5}\right)^{2-3 x}$
$\Rightarrow \left(\frac{3}{5}\right)^{-\frac{1}{2}}=\left(\frac{3}{5}\right)^{2-3 x}$
$\Rightarrow -\frac{1}{2}=2-3 x$
$\Rightarrow -1=4-6 x$
$\Rightarrow -5=-6 x$
$\Rightarrow x=\frac{5}{6}$
View full question & answer→Question 43 Marks
Solve for $x:9 \times 81^x=\frac{1}{27^{x-3}}$
Answer$9 \times 81^x=\frac{1}{27^{x-3}}$
$ \Rightarrow 3^2 \times 3^{4 x}=\frac{1}{3^3(x-3)}$
$\Rightarrow 3^2 \times 3^{4 x}=\frac{1}{3^{3 x-9}}$
$ \Rightarrow 3^2 \times 3^{4 x} \times 3^{3 x-9}=1 $
$ \Rightarrow 3^{2+4+3 x-9}=1 \times 3^0$
$\Rightarrow 2+4+3 x-9=0$
$ \Rightarrow 3 x-3=0 $
$ \Rightarrow x=1$
View full question & answer→Question 53 Marks
Solve for $x :p ^{-5}=\frac{1}{ p ^{x+1}}$
Answer$ p ^{-5}=\frac{1}{ p ^{x+1}}$
$ \Rightarrow p ^{-5} \times p ^{x+1}=1$
$ \Rightarrow p ^{-5+x+1}=1$
$\Rightarrow p ^{x-4}= p ^0 $
$ \Rightarrow x -4=0 $
$ \Rightarrow x =4$
View full question & answer→Question 63 Marks
Solve for $x:3 \times 7^x = 7 \times 3^x$
Answer$3 \times 7^x=7 \times 3^x$
$ \Rightarrow \frac{7^x}{7}=\frac{3^x}{3}$
$\Rightarrow 7^{x-1}=3^{x-1} \ldots \ldots ($Using $a^m \div a^n=a^{m-n}) $
$ \Rightarrow 7^{x-1}=3^{x-1} \times 1 $
$ \Rightarrow 7^{x-1}=3^{x-1} \times 7^0 \ldots \ldots($Using $a^0=1)$
$ \Rightarrow x-1=0 $
$ \Rightarrow x=1$
View full question & answer→Question 73 Marks
Solve for $x:2^{2x+1}= 8$
Answer$2^{2x+1}= 8$
$\Rightarrow 2^{2x+1}= 2^3$
$\Rightarrow 2x + 1 = 3$
$\Rightarrow 2x = 2$
$\Rightarrow x = 1.$
View full question & answer→Question 83 Marks
Simplify the following:$\frac{2^{ m } \times 3-2^{ m }}{2^{ m +4}-2^{ m +1}}$
Answer$\frac{2^{ m } \times 3-2^{ m }}{2^{ m +4}-2^{ m +1}}$
$=\frac{2^{ m }(3-1)}{2^{ m }\left(2^4-2\right)} $
$ =\frac{2}{16-2} $
$ =\frac{2}{14} $
$=\frac{1}{7} .$
View full question & answer→Question 93 Marks
Simplify the following:$\frac{3^{x+1}+3^x}{3^{x+3}-3^{x+1}}$
Answer$ \frac{3^{x+1}+3^x}{3^{x+3}-3^{x+1}}$
$= \frac{3^x(3+1)}{3^x\left(3^3-3\right)}$
$= \frac{4}{27-3}$
$= \frac{4}{24}$
$= \frac{1}{6} .$
View full question & answer→Question 103 Marks
Simplify the following:$\frac{5^x \times 7-5^x}{5^{x+2}-5^{x+1}}$
Answer$\frac{5^x \times 7-5^x}{5^{x+2}-5^{x+1}} $
$=\frac{5^x(7-1)}{5^{x+1}(5-1)} $
$ =\frac{5^{x-x-1} \times 6}{4} $
$=\frac{5^{-1} \times 6}{6^4}$
$=\frac{6^4}{5 \times 4}$
$ =\frac{3}{10} .$
View full question & answer→Question 113 Marks
Simplify the following:$x^{ m +2 n } \cdot x^{3 m -8 n } \div x^{5 m -60}$
Answer$x^{m+2 n} \cdot x^{3 m-8 n} \div x^{5 m-60}$
$ =x^{m+2 n+3 m-8 n-5 m-(-60)} \ldots .($ Using $a^m \times a^n=a^{m+n}$ and $a^m \div a^n=a^{m-n})$
$=x^{m+2 n+3 n-8 n-5 n+60}$
$=x^{-m-6 n+60}$
View full question & answer→Question 123 Marks
Simplify the following:$\left(8 x^6 y^3\right)^{\frac{2}{3}}$
Answer$\left(8 x^6 y^3\right)^{\frac{2}{3}}$
$ =\left(2^3 x^6 y^3\right)^{\frac{2}{3}}$
$ =\left(2^3\right)^{\frac{2}{3}}\left(x^6\right)^{\frac{2}{3}}\left(y^3\right)^{\frac{2}{3}} \ldots \ldots($ Using $(a \times b)^n=a^n \times b^n)$
$ =(2)^{3 \times \frac{2}{3}}(x)^{6 \times \frac{2}{3}}(y)^{3 \times \frac{2}{3}} \ldots . .($ Using $\left(a^m\right)^n=a^{m n})$
$=(2)^2(x)^4(y)^2 $
$ =4 x^4 y^2$
View full question & answer→Question 133 Marks
Simplify the following:$\left(27 \times x^9\right)^{\frac{2}{3}}$
Answer$\left(27 \times x{^ 9}\right)^{\frac{2}{3}}$
$=\left(3^3 \times x^9\right)^{\frac{2}{3}} $
$ =\left(3^3\right)^{\frac{2}{3}}\left(x^9\right)^{\frac{2}{3}} \ldots \ldots($ Using $( a \times b )^n=a^n \times b^n) $
$=(3)^{3 \times \frac{2}{3}}(\times)^{9 \times \frac{2}{3}} \ldots . .($ Using $\left(a^m\right)^n=a^{m n}) $
$ =(3)^2 x^{3\times 2}$
$=9 x^6 .$
View full question & answer→Question 143 Marks
Evaluate the following:$16^{\frac{3}{4}}+2\left(\frac{1}{2}\right)^{-1} \times 3^0$
Answer$16^{\frac{3}{4}}+2\left(\frac{1}{2}\right)^{-1} \times 3^0$
$=2^{4 \times \frac{3}{4}}+2 \times 2 \times 1$
$ =2^3+4 $
$=2^3+4$
$=8+4$
$ =12$
View full question & answer→Question 153 Marks
Evaluate the following:$9^{\frac{5}{2}}-3 \times 5^0-\left(\frac{1}{81}\right)^{\frac{-1}{2}}$
Answer$9^{\frac{5}{2}}-3 \times 5^0-\left(\frac{1}{81}\right)^{\frac{-1}{2}}$
$=3^{2 \times \frac{5}{2}}-3 \times 1-\left(\frac{1}{81}\right)^{\frac{-1}{2}} $
$=3^5-3-9^{2 \times \frac{1}{2}} $
$=243-3-9$
$ =231 .$
View full question & answer→Question 163 Marks
Evaluate the following:$\frac{4^3 \times 3^7 \times 5^6}{5^8 \times 2^7 \times 3^3}$
Answer$ \frac{4^3 \times 3^7 \times 5^6}{5^8 \times 2^7 \times 3^3}$
$= \frac{\left(2^2\right)^3 \times 3^{7-3}}{5^{8-6} \times 2^7}$
$= \frac{2^6 \times 3^4}{5^2 \times 2^7}$
$= \frac{3^4}{5^2 \times 2^{7-6}}$
$= \frac{81}{5^2 \times 2^1}$
$= \frac{81}{50} .$
View full question & answer→Question 173 Marks
Write each of the following in the simplest form:$(b^{-2} - a^{-2}) \div (b^{-1} - a^{-1})$
Answer$\left(b^{-2}-a^{-2}\right) \div\left(b^{-1}-a^{-1}\right) $
$=a^{-3+2+0} \ldots \ldots($ Using $a^m \times a^n=a^{m+n}) $
$ =a^{-1} $
$ =\frac{1}{a} .$
View full question & answer→Question 183 Marks
Write each of the following in the simplest form:$a^{-3}\times a^2 \times a^0$
Answer$a^{-3} \times a^2 \times a^0 $
$ =a^{-3+2+0} \ldots . . .($ Using $a^m \times a^n=a^{m-n}) $
$ =a^{-1} $
$=\frac{1}{a}$
View full question & answer→Question 193 Marks
Write each of the following in the simplest form:$a^{\frac{1}{3}} \div a^{-\frac{2}{3}}$
Answer$a^{\frac{1}{3}} \div a^{-\frac{2}{3}} $
$=a^{\frac{1}{3}-\left(-\frac{2}{3}\right)} \ldots . .($ Using $a^m \div a^n=a^{m-n})$
$ =a^{\frac{1}{3}+\frac{2}{3}} $
$ =a^{\frac{1+2}{3}} $
$ =a^1 $
$=a .$
View full question & answer→Question 203 Marks
Write each of the following in the simplest form:$(a^3)^5 \times a^4$
Answer$(a^3)^5 \times a^4$
$= (a)^{3\times5} \times a^4 .....($Using $(a^m)^n = a^{mn})$
$= (a)^{15} \times a^4$
$= a^{15 +4} .....($Using $a^m \times a^n = a^{m +n})$
$= a^{19}.$
View full question & answer→Question 213 Marks
If $2^x=3^y=12^z$; show that $\frac{1}{z}=\frac{1}{y}+\frac{2}{x}$.
AnswerLet $2^x=3^y=12^z= k $
$\Rightarrow 2= k ^{\frac{1}{x}}, 3= k ^{\frac{1}{y}}, 12= k ^{\frac{1}{x}}$
Now, $12=2 \times 2 \times 3$
$\Rightarrow k ^{\frac{1}{x}}= k ^{\frac{1}{x}} \times k ^{\frac{1}{x}} \times k ^{\frac{1}{y}} $
$\Rightarrow \frac{1}{z}=\frac{1}{x}+\frac{1}{x}+\frac{1}{y} $
$\Rightarrow \frac{1}{z}=\frac{2}{x}+\frac{1}{y}$
View full question & answer→Question 223 Marks
If $2400=2^x \times 3^y \times 5^z$, find the numerical value of $x, y, z$. Find the value of $2^{-x} \times 3^y \times 5^z$ as fraction.
Answer$2400=2^x \times 3^y \times 5^z$
$2400=2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 $
$\therefore 2^x \times 3^y \times 5^z=2^5 \times 3^1 \times 5^2 $
$ \Rightarrow x=5, y=1, z=2 $
$ \therefore 2^{-x} \times 3^y \times 5^z=2^{-5} \times 3^1 \times 5^2$
$=\frac{1}{32} \times 3 \times 25$
$=\frac{75}{32} .$
View full question & answer→Question 233 Marks
If $2250=2^a \cdot 3^b \cdot 5^c$, find $a, b$ and $c$. Hence, calculate the value of $3^a \times 2^{-b} \times 5^{-c}$.
AnswerGiven $2250=2^a \cdot 3^b \cdot 5^c $
$\Rightarrow 3^2 \times 5^3 \times 2=2^a \cdot 3^b \cdot 5 $
$ \Rightarrow a=1, b=2, c=3 $
$ 3^a \times 2^{-b} \times 5^{-x} $
$ =3^1 \times 2^{-2} \times 5^{-3} $
$ =\frac{3}{2^2 \times 5^3} $
$ =\frac{3}{500} .$
View full question & answer→Question 243 Marks
If $a ^{ x }= b ^{ y }= c ^{ z }$ and $b ^2= ac$, prove that $y =\frac{2 \times z}{z+x}$
AnswerLet $a ^{ x }= b ^{ y }= c ^{ z }= k $
$\Rightarrow a = k ^{\frac{1}{x}}, b = k ^{\frac{1}{y}}, c = k ^{\frac{1}{2}}$
It is also given that $b^2=a c$
$\Rightarrow k ^{\frac{2}{y}}= k ^{\frac{1}{x}} \times k ^{\frac{1}{2}} $
$\Rightarrow k ^{\frac{2}{y}}= k ^{\frac{1}{x}+\frac{1}{x}} $
$\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z} $
$\Rightarrow y =\frac{2 \times z}{z+x} .$
View full question & answer→Question 253 Marks
If $\sqrt[x]{ a }=\sqrt[y]{ b }=\sqrt[z]{ c }$ and $abc =1$, prove that $x + y + z =0$
AnswerLet $\sqrt[x]{a}=\sqrt[x]{b}=\sqrt[x]{c} $
$\Rightarrow a^{\frac{1}{x}}=k, b^{\frac{1}{y}}=k, c^{\frac{1}{x}}=k $
$\Rightarrow a=k, b=k, c=k$
It is also given that abc $=1$
$\Rightarrow k ^{ x } \times k ^y \times k ^z=1 $
$\Rightarrow k ^{x+y+z}= k ^{\cdot} $
$\Rightarrow x + y + z =0 .$
View full question & answer→Question 263 Marks
Evaluate the following: $\left(2 \frac{10}{27}\right)^{\frac{2}{3}}$
Answer$\left(2 \frac{10}{27}\right)^{\frac{2}{3}}$
$=\left(\frac{64}{27}\right)^{\frac{2}{3}} $
$ =\left(\frac{4}{3}\right)^{3 \times \frac{2}{3}} $
$ =\left(\frac{4}{3}\right)^2$
$=\frac{16}{9} .$
View full question & answer→Question 273 Marks
Evaluate the following: $(0.00243)^{-\frac{3}{5}}$
Answer$(0.00243)^{-\frac{3}{5}}$
$=\frac{1}{(0.00243)^{\frac{3}{5}}}$
$ =\frac{1}{\left(0.3^5\right)^{\frac{3}{5}}} $
$=\frac{1}{(0.3)^3}$
$=\frac{1}{0.027} .$
View full question & answer→Question 283 Marks
Find the value of $k$ in each of the following:$(\sqrt{9})^{-7} \times(\sqrt{3})^{-5}=3^k$
Answer$(\sqrt{9})^{-7} \times(\sqrt{3})^{-5}=3^k$
$\Rightarrow\left\{\left(3^2\right)^{\frac{1}{2}}\right\}^{-7}\left\{(3)^{\frac{1}{2}}\right\}^{-5}=3^k$
$\Rightarrow 3^{-7} \times 3^{\frac{-5}{2}}=3^k$
$\Rightarrow 3^{-7-\frac{5}{2}}=3^k$
$\Rightarrow 3^{\frac{-145}{2}}=3^k$
$\Rightarrow 3^{\frac{-19}{2}}=3^k$
$\Rightarrow k=\frac{-19}{2} .$
View full question & answer→Question 293 Marks
Find the value of $k$ in each of the following:$\sqrt[4]{\sqrt[3]{x^2}}=x^k$
Answer$\sqrt[4]{\sqrt[3]{x^2}}= x ^{ k }$
$\Rightarrow\left\{\left(x^2\right)^{\frac{1}{3}}\right\}^{\frac{1}{4}}= x ^{ k } $
$ \Rightarrow\left(x^2\right)^{\frac{1}{12}}= x ^{ k } $
$ \Rightarrow x^{\frac{2}{12}}= x ^{ k } $
$ \Rightarrow x^{\frac{1}{6}}= x ^{ k }$
$ \Rightarrow k =\frac{1}{6} .$
View full question & answer→Question 303 Marks
Find the value of k in each of the following:$(\sqrt[3]{8})^{\frac{-1}{2}}=2^k$
Answer$(\sqrt[3]{8})^{\frac{-1}{2}}=2^k $
$ \Rightarrow 8^{\frac{1}{3} \times \frac{-1}{2}}=2^k$
$\Rightarrow\left(2^3\right)^{\frac{1}{3} \times \frac{-1}{2}}=2^k $
$ \Rightarrow\left(2^3\right)^{\frac{1}{3} \times \frac{-1}{2}}=2^k$
$ \Rightarrow 2^{\frac{-1}{2}}=2^k $
$\Rightarrow k =-\frac{1}{2}$
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