Question 12 Marks
Express the number appearing in the statement in standard form :Thickness of a thick paper is $0.07\ mm$.
Answer$0.07$
$ = \frac{7}{{100}}$ mm
$ = \frac{7}{{{{10}^2}}}$ mm
$= 7$ $\times$ $10^{-2}$ mm
View full question & answer→Question 22 Marks
Express the number appearing in the statement in standard form: The size of a plant cell is $0.00001275\ m$.
Answer$0.00001275$
$= 0.00001275$$\times \frac{10^{5}}{10^{5}}=1.275 \times 10^{-5} $
The size of a plant cell is $1.275 \times 10^{-5} $ m
View full question & answer→Question 32 Marks
Express the number appearing in the statement in standard form : Size of a bacteria is $0.0000005\ m$.
Answer$0.0000005\ m$
$ = \frac{5}{{10000000}}m$
$ = \frac{5}{{{5^7}}}m$
$= 5$ $\times$ $10^{-7}$ $m$
View full question & answer→Question 42 Marks
Express the number appearing in the statement in standard form :Charge of an electron is $0.00000000000000000016$ coulomb.
Answer$0.00000000000000000016$
$\frac{{16}}{{100000000000000000000}}$
$ = \frac{{16}}{{{{10}^{20}}}}$
$ = \frac{{1.6}}{{{{10}^{19}}}}$
$= 1.6$ $\times$ $10–19$
View full question & answer→Question 52 Marks
Express the number appearing in the statement in standard form :$1$ micron is equal to $ = \frac{1}{{1000000}}m$.
Answer$ = \frac{1}{{1000000}}m$
$ = \frac{1}{{{{10}^6}}}m$
$= 1$ $\times$ $10^{-6}m.$
View full question & answer→Question 62 Marks
Express $3.61492$ $\times$ $10^6$ in usual form.
Answer$3.61492$ $\times$ $10^6$
$= 3.61492$ $\times$ $1000000$
$= 3614920$
View full question & answer→Question 72 Marks
Express $5.8$ $\times$ $10^{12}$ in usual form.
Answer$5.8$ $\times$ $10^{12}$
$= 5.8$ $\times$ $1000000000000$
$= 5800000000000$
View full question & answer→Question 82 Marks
Express $1.0001$ $\times$ $10^9$ in usual form.
Answer$1.0001$ $\times$ $10^9$
$= 1.0001$ $\times$ $1000000000$
$= 1000100000$
View full question & answer→Question 92 Marks
Express $3 \times 10^{-8}$ in usual form.
AnswerWe have, $3 x 10^{-8}$
$= 0.00000003$
View full question & answer→Question 102 Marks
Express $4.5 \times 10^4$ in usual form.
Answer$4.5 \times 10^4$
$= 4.5$ $\times$ $10000$
$= 45000$
View full question & answer→Question 112 Marks
Express $3.02$ $\times$ $10^{-6}$ in usual form.
Answer$3.02$ $\times$ $10^{-6}$
$ = \frac{{3.02}}{{{{10}^6}}}$
$ = \frac{{3.02}}{{1000000}}$
$= 0.00000302$
View full question & answer→Question 122 Marks
Express the $31860000000$ in standard form.
Answer$31860000000$
$= 31860000000$ $\times \frac{10^{10}}{10^{10}}=3.186 \times 10^{10}$
View full question & answer→Question 132 Marks
Express the $0.00000000837$ in standard form.
Answer$0.00000000837 = 0.00000000837$ $\times \frac{10^{9}}{10^{9}}=8.37 \times 10^{-9}$
View full question & answer→Question 142 Marks
Express the $6020000000000000$ in standard form.
Answer$6020000000000000$
$= 6020000000000000$$\times \frac{10^{15}}{10^{15}}=6.02 \times 10^{15}$
View full question & answer→Question 152 Marks
Express $0.00000000000942$ in standard form.
Answer$0.00000000000942$ $ = \frac{{942}}{{{{10}^{14}}}}$
$ = \frac{{9.42}}{{{{10}^{12}}}}$
$= 9.42 × 10^{-12}$
View full question & answer→Question 162 Marks
Express the number $0.0000000000085$ in standard form.
Answer$0.0000000000085$
$= 0.0000000000085$ $\times\frac{10^{12}}{10^{12}}=8.5 \times 10^{-12}$
View full question & answer→Question 172 Marks
Evaluate :${\left\{ {{{\left( {\frac{1}{3}} \right)}^{ - 1}} - {{\left( {\frac{1}{4}} \right)}^{ - 1}}} \right\}^{ - 1}}$
Answer${\left\{ {{{\left( {\frac{1}{3}} \right)}^{ - 1}} - {{\left( {\frac{1}{4}} \right)}^{ - 1}}} \right\}^{ - 1}}$
$ = {\left( {\frac{{{1^{ - 1}}}}{{{3^{ - 1}}}} - \frac{{{1^{ - 1}}}}{{{4^{ - 1}}}}} \right)^{ - 1}} = {\left( {\frac{{{3^1}}}{{{1^1}}} - \frac{{{4^1}}}{{{1^1}}}} \right)^{ - 1}}$
$ = {\left( {\frac{3}{1} - \frac{4}{1}} \right)^{ - 1}}$ $= (3 – 4)^{-1}$
$ = {( - 1)^{ - 1}} = \frac{1}{{{{( - 1)}^1}}}$
$ = \frac{1}{{( - 1)}} = - 1$
View full question & answer→Question 182 Marks
Find the value of m for which $5^m$ $\div$ $5^{-3}= 5^5$
Answer$5^m$ $\div$ $5^{-3}= 5^5$
$\therefore \frac{{{5^m}}}{{{5^{ - 3}}}} = {5^5}$
$\therefore$ $ 5^{m-(-3)}= 5^5 $
$\therefore$ $ 5^{m+3}= 5^5 $
$\therefore$ $m + 3 = 5$ [When the bases are same , powers are equal]
$\therefore$ $m = 5 – 3$
$\therefore$ $m = 2$
View full question & answer→Question 192 Marks
Evaluate:$\left(5^{-1} \times 2^{-1}\right) \times 6^{-1}$
Answer$\left(5^{-1} \times 2^{-1}\right) \times 6^{-1}$
$ = \left( {\frac{1}{5} \times \frac{1}{2}} \right) \times \frac{1}{6}$
$ = \frac{1}{{10}} \times \frac{1}{6}$
$ = \frac{1}{{60}}$
View full question & answer→Question 202 Marks
Evaluate: $\frac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}$
Answer$\frac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}$$ = \frac{{{8^{ - 1}} \times (5 \times 5 \times 5)}}{{{2^{ - 4}}}}$
$ = \frac{1}{8} \times {2^4} \times 125$
$ = \frac{1}{8} \times 2 \times 2 \times 2 \times 2 \times 125$
$ = 2 \times 125 = 250$
View full question & answer→Question 212 Marks
Find the value of :$[3^{-1}+ 4^{-1}+ 5^{-1}]^0$
Answer$[3^{-1}+4^{-1}+ 5^{-1}]^0$
$ = {\left[ {\frac{1}{3} + \frac{1}{4} + \frac{1}{5}} \right]^0}$
$ = {\left( {\frac{{20 + 15 + 12}}{{60}}} \right)^0}$
$ = {\left( {\frac{{47}}{{60}}} \right)^0}$
= 1
View full question & answer→Question 222 Marks
Find the value of: $(3^0+ 4^{-1})$ $\times$ $2^2$
Answer$(3^0+ 4^{-1})$ $\times$ $2^2$
$ = \left( {1 + \frac{1}{4}} \right) \times 4$
$ = \frac{5}{4} \times 4$
= 5
View full question & answer→Question 232 Marks
Simplify and express the result in power notation with positive exponent. $2^{-3}× (– 7)^{-3}$
Answer$2^{-3}× (– 7)^{-3}$
$ = \frac{1}{{{2^3}}} \times \frac{1}{{{{( - 7)}^3}}}$
$ = \frac{1}{{{{\left[ {2 \times ( - 7)} \right]}^3}}}$
$ = \frac{1}{{{{( - 14)}^3}}}$
View full question & answer→Question 242 Marks
Simplify and express the result in power notation with positive exponent: $\left(3^{-7} \div 3^{-10}\right) \times 3^{-5}$
Answer$\left(3^{-7} \div 3^{-10}\right) \times 3^{-5}$
$ = \left( {\frac{{{3^{ - 7}}}}{{{3^{ - 10}}}}} \right) \times \frac{1}{{{3^5}}}$
$ = {3^{( - 7) - ( - 10)}} \times {3^{\frac{1}{5}}}$
$ = {3^{ - 7 + 10}} \times {3^{\frac{1}{5}}}$
$ = \frac{{{3^3}}}{{{3^5}}}$
$ = \frac{1}{{{3^{5 - 3}}}}$
$ = \frac{1}{{{{(3)}^2}}}$
View full question & answer→Question 252 Marks
Simplify and express the result in power notation with positive exponent: ${( - 3)^4} \times {\left( {\frac{5}{3}} \right)^4}$
Answer${( - 3)^4} \times {\left( {\frac{5}{3}} \right)^4}$
$ = {\{ ( - 1) \times 3\} ^4} \times {\left( {\frac{5}{3}} \right)^4}$
$ = {( - 1)^4} \times {3^4} \times \frac{{{5^4}}}{{{3^4}}}$
$= (5)^4$
View full question & answer→Question 262 Marks
Simplify and express the result in power notation with positive exponent: ${\left( {\frac{1}{{{2^3}}}} \right)^2}$
Answer${\left( {\frac{1}{{{2^3}}}} \right)^2}$
$ = \frac{{{1^2}}}{{{{({2^3})}^2}}}$
$ = \frac{1}{{{2^{3 \times 2}}}}$
$ = \frac{1}{{{2^6}}}$
View full question & answer→Question 272 Marks
Simplify and express the result in power notation with positive exponent.$(– 4)^5÷ (– 4)^8$
Answer$(– 4)^5$ $\therefore$ $(– 4)^8$
$ = \frac{{{{( - 4)}^5}}}{{{{( - 4)}^8}}}$
$ = \frac{1}{{{{( - 4)}^{8 - 5}}}}$
$ = \frac{1}{{{{( - 4)}^3}}}$
View full question & answer→Question 282 Marks
Evaluate : ${\left( {\frac{1}{2}} \right)^{ - 5}}$
Answer${\left( {\frac{1}{2}} \right)^{ - 5}}$
$ = \frac{1}{{{{\left( {\frac{1}{2}} \right)}^5}}}$
$ = \frac{1}{{\frac{{{1^5}}}{{{2^5}}}}}$
$ = \frac{1}{{\left( {\frac{1}{{32}}} \right)}}$
= 32
View full question & answer→Question 292 Marks
Evaluate :$(–4)^{-2}$
Answer$(–4)^{-2}$
$ = \frac{1}{{{{( - 4)}^2}}} = \frac{1}{{16}}$
View full question & answer→Question 302 Marks
Evaluate :$3^{-2}$
Answer$3^{-2}$
$ = \frac{1}{{{3^2}}} = \frac{1}{9}$
View full question & answer→Question 312 Marks
Express the number $3.52$ $\times$ $10^5$ in the usual form.
AnswerWe have, $3.52$ $\times$ $10^5$
$= 3.52$ $\times$ $100000$
$= 352000$
View full question & answer→Question 322 Marks
Simplify: $\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-5}$
Answer$\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-5}$ = $\frac{5^{-7}}{8^{-7}} \times \frac{8^{-5}}{5^{-5}}=\frac{5^{-7}}{5^{-5}} \times \frac{8^{-5}}{8^{-7}}$
= $5^{(-7)-(-5)} \times 8^{(-5)-(-7)}$ = $5^{-2} \times 8^{2}=\frac{8^{2}}{5^{2}}=\frac{64}{25}$
View full question & answer→Question 332 Marks
Simplify: $\left\{\left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-2}$
Answer$\left\{\left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-2}$
= $\left\{\frac{1^{-2}}{3^{-2}}-\frac{1^{-3}}{2^{-3}}\right\} \div \frac{1^{-2}}{4^{-2}}$
= $\left\{\frac{3^{2}}{1^{2}}-\frac{2^{3}}{1^{3}}\right\} \div \frac{4^{2}}{1^{2}}=\{9-8\} \div 16=\frac{1}{16}$
View full question & answer→Question 342 Marks
Find m so that $(–3)^{m+1} × (–3)^{5} = (–3)^{7}$
Answer$(–3)^{m+1} × (–3)^{5} = (–3)^{7}$
$(-3)^{m+1+5}= (-3)^7$
$(-3)^{m+6}= (-3)^7$
Therefore, $m + 6 = 7$
or $m = 7 – 6 = 1$
View full question & answer→Question 352 Marks
Simplify the exponential form: $\frac{1}{8} \times(3)^{-3}$
Answer$\frac{1}{8} \times(3)^{-3}$
= $\frac{1}{2^{3}} \times(3)^{-3}=2^{-3} \times 3^{-3}$
= $(2 \times 3)^{-3}=6^{-3}=\frac{1}{6^{3}}$
View full question & answer→Question 362 Marks
Simplify the exponential form: $(-4)^{-3} \times(5)^{-3} \times(-5)^{-3}$
Answer$(-4)^{-3} \times(5)^{-3} \times(-5)^{-3}$
= $[(-4) \times 5 \times(-5)]^{-3}=[100]^{-3}=\frac{1}{100^{3}}$
View full question & answer→Question 372 Marks
Simplify the exponential: $(2^{5} ÷ 2^{8} )^{5} × 2^{– 5} $
Answer$(2^{5} ÷ 2^{8} )^{5} \times 2^{– 5} $
=$\left(2^{5-8}\right)^{5} \times 2^{-5}$
= $\left(2^{-3}\right)^{5} \times 2^{-5}=2^{-15-5}=2^{-20}=\frac{1}{2^{20}}$
View full question & answer→Question 382 Marks
Express $4^{-3}$ as a power with the base $2$.
AnswerWe have, $4 = 2 × 2 = 2^{2}$
Therefore, $(4)^{-3}= (2^2)^{-3}= 2^2$ $\times$$(– 3)$
[because $(am)n= an$$\times$$m$ ]
$= 2^{-6}$
View full question & answer→Question 392 Marks
Simplify $2^5 \div 2^{-6}$
Answerwe have, $2^{5} \div 2^{-6}$
= $2^{5-(-6)}$$\quad\left[a^{m} \div a^{n}=a^{m-n}\right]$
= $2^{11}$
View full question & answer→Question 402 Marks
Simplify: $(– 4)^{5} × (– 4)^{-10}$
Answer$(– 4)^{5} × (– 4)^{-10}$
= $(– 4)^{(5 – 10)} $ [because $a^{m} × a^{n} = a^{m + n}$]
= $ (– 4)^{–5}$
= $\frac{1}{(-4)^{5}}$ [because $ a^{-m}=\frac{1}{a^{m}}$]
= $\frac {-1}{1024}$
View full question & answer→Question 412 Marks
Simplify the exponential form: $(-3)^{4} \times\left(\frac{5}{3}\right)^{4}$
Answer\begin{aligned} (-3)^4 \times\left(\frac{5}{3}\right)^4 & =(-1 \times 3)^4 \times \frac{5^4}{3^4}=(-1)^4 \times 3^4 \times \frac{5^4}{3^4} \\ & =(-1)^4 \times 5^4=5^4 \quad\left[(-1)^4=1\right]\end{aligned}
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