1.6 \(\times\) 10^-1

\(\left\{\left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-2} =\left\{\frac{1^{-2}}{3^{-2}}-\frac{1^{-3}}{2^{-3}}\right\} \div \frac{1^{-2}}{4^{-2}} \)
\(\\ =\left\{\frac{3^{2}}{1^{2}}-\frac{2^{3}}{1^{3}}\right\} \div \frac{4^{2}}{1^{2}}=\{9-8\} \div 16=\frac{1}{16} \)

2⁵ ÷ 2^{– 6} = 2^{5 – (– 6}) = 2¹¹

3.5 \(\times\) 10^-4

7 \(\times\) 10^-6

3.72 x 10⁹ kg

5.68 x 10^-4

We have, 0.00000000000942
=\(\frac{942}{10^{14}}=\frac{9.42\times 100}{10^{14}}=9.42\times 10^{2}\times 10^{-14} \quad [\because \frac{1}{a^{m}}=a^{-m}]\)
= \(9.42\times 10^{2-14}=9.42\times10^{-12}\quad [\because a^{m}\times a^{n}=a^{m+n}]\)
which is the required standard form.

We have , {\((\frac{1}{3})^{-1}-(\frac{1}{4})^{-1}\)}^-1
\(\left\{ \frac { { \left( 1 \right) }^{ -1 } }{ { \left( 3 \right) }^{ -1 } } -\frac { { \left( 1 \right) }^{ -1 } }{ { \left( 4 \right) }^{ -1 } } \right\} ^{ -1 }\quad \quad \quad \left[ \because \quad \left( \frac { a }{ b } \right) ^{ m }=\frac { { a }^{ m } }{ { b }^{ m } } \quad \right] \)
\(=\left\{ \frac { 3 }{ 1 } -\frac { 4 }{ 1 } \right\} ^{ -1 }\quad \quad \quad \left[ \because \quad { a }^{ -m }=\frac { 1 }{ { a }^{ m } } \right] \)
= (3-4)^-1 = (-1)^-1=\(\frac{1}{(-1)^{1}}\)     \( \left[ \because \quad { a }^{ -m }=\frac { 1 }{ { a }^{ m } } \right] \)
=\(\frac{1}{-1}=-1\)

Multiplicative inverse of 10^-100 is 10¹⁰⁰.

34500000 - 345 \(\times\) 100000
= 3.45 \(\times\) 100 \(\times\) 100000
= 3.45 \(\times\) 10² \(\times\) 10⁵
= 3.45 \(\times\) 10²⁺⁵
= 3.45 \(\times\) 10⁷
Thus, 34500000 = 3.45 \(\times\) 10⁷

Since \({ \left( \frac { 1 }{ 4 } \right) }^{ -2 }={ \left( \frac { 4 }{ 1 } \right) }^{ 2 }={ (4) }^{ 2 }=16\)
\({ \left( \frac { 1 }{ 3 } \right) }^{ -3 }={ (3) }^{ 3 }=3\times 3\times 3=27\)
\({ \left( \frac { 1 }{ 2 } \right) }^{ -4 }={ (2) }^{ 4 }=2\times 2\times 2\times 2=16\)
\(\therefore \quad { \left( \frac { 1 }{ 4 } \right) }^{ -2 }+{ \left( \frac { 1 }{ 3 } \right) }^{ -3 }+{ \left( \frac { 1 }{ 2 } \right) }^{ -4 }=16+27+16=59\)

Size of red blood cell = 0.000007 m
\(=\frac { 7 }{ 1000000 } =7\times { 10 }^{ -6 }m\)
Size of the plant cell = 0.0000129 m
\(=\frac { 129 }{ 10000000 } m=\frac { 1.29\times { 10 }^{ 2 } }{ 10000000 } m=1.29\times { 10 }^{ -5 }m\)
Now \(\frac { 7\times { 10 }^{ -6 } }{ 1.29\times { 10 }^{ -5 } } =\frac { 7\times { 10 }^{ -1 } }{ 1.29 } \)
\(=\frac { 7\times { 10 }^{ -1 } }{ 1.3 } =\frac { 0.7 }{ 1.3 } =\frac { 1 }{ 2 } \) (approx.)
Thus, the size of a red blood cell is half of the plant cell size.

The cells of a bacteria double itself every hour, i.e.
the cells of a bacteria after 1 h = 2
∴  The cells of a bacteria after 2 h = 2 x 2 = 2²
The cells of a bacteria after 3 h = 2 x 2² = 2³
The cells of a bacteria after 12 h = 2¹²

0.00015

2

-45

(a) Generation table (according to the diagram) showing the number of ancestors are given below:

(b) Equation for the number of ancestors in generation n is 2ⁿ [according to the part (a)].

The fraction of substance remains after 7 half- lives =3^-7=\(\frac{1}{3^{7}}\)
ஃ \(3^{-x}=\frac{1}{243}=\frac{1}{3\times3\times3\times3\times3}=\frac{1}{3^{5}}\)
⇒ \(3^{-x}=3^{-5}\)
⇒ \((3)^{-x}=(3)^{-5}\)
⇒ -x=-5  [∵ bases are same]]
Hence, after 5 half-lives, the fraction will be \(\frac{1}{243}\) of the original.

(a) Mass of each celestial body in standard in notation is given below.
Mass of Sun = 1.99 x 10³⁰ kg
Mass of Mercury = 3.3 x 10²³ kg
Mass of Venus = 4.870 x 10²⁴ kg
Mass of Earth = 5.970 x 10²⁴ kg
Mass of Mars = 6.42 x 10²⁹ kg
Mass of Jupiter = 1.9 x 10²⁷ kg
Mass of Saturn = 5.68 x10²⁶ kg
Mass of Uranus = 8.68 x 10²⁵ kg
Mass of Neptune = 1.02 x 10²⁶ kg
Mass of Pluto = 1.27 x 10²² kg
Mass of Moon = 7.35 x 10²² kg
(b) On observing the masses of the all given planets and the Moon in the exponent form, we see that the order of planets and the Moon by mass from least to greatest is Pluto < Moon < Mercury < Venus < Earth < Uranus < Neptune < Saturn < Jupiter < Mars
(c) ஃ Mass of Earth = 5.970 X10²⁴ kg and mass of Venus = 4.870 x 10²⁴ kg ,
Thus, we can say that planet Venus has about same mass as the Earth.

2.1 x 10^-6= 0.0000021

(-1)^{odd natural number} = -1

\((-2)^{-2}={1\over (-2)^2}={1\over 4}\)

\(10^{-1}={1\over 10^1}={1\over 10}\)