1.6 \(\times\) 10^-1

\(\frac { 1 }{ -5 } \)

3.52 x 10⁵ = 3.52 x 100000 = 352000

4.05 \(\times\) 10⁶

x = 2

∵ x=(100)^1-4 ÷ (100)⁰=(100)^-3    [∵ (100)⁰=1]
∴ \(x^{-3}=\frac{1}{x^{3}}=\frac { 1 }{ \left\{ { \left( 100 \right) }^{ -3 } \right\} ^{ 3 } } =\frac{1}{100^{-9}}\)      [∵ (a^m)ⁿ=a^mn]
= 100⁹

∵ \(2^{-1}\times 3^{-1}=\frac{1}{2}\times \frac{1}{3}=\frac{1}{6}\)
∴ \([2^{-1}\times 3^{-1}]^{-1}=[\frac{1}{6}]^{-1}=6\)

Here, 0.00001275 = \(\frac{1275}{100000000}\)
= \(\frac{1275}{10^{8}}=\frac{1.275 \times 1000}{10^{8}}=1.275\times 10^{3}\times 10^{-8}\)  \([\because \frac{1}{a^{m}}=a^{-m}]\)
= 1.275 x 10^3-8 = 1.275 x 10^-5m       [∵ a^mxaⁿ = a^m+n]
Hence, size of a plant cell is 1.275 x 10^-5 m.

Here, 0.0000005 = \(\frac{5}{10000000}=\frac{5}{10^{7}}=5\times 10^{-7}m\)     \([\because \frac{1}{a^{m}}=a^{-m}]\)
Hence, size of a bacteria is 5 x 10^-7 m.

\(\frac{4097}{64}\)

10

\({ \left( \frac { 5 }{ 8 } \right) }^{ -7 }\times { \left( \frac { 8 }{ 5 } \right) }^{ -4 }\)  \(\because \quad { a }^{ -m }=\frac { 1 }{ { a }^{ m } } \)
\(\because \quad { \left( \frac { 8 }{ 5 } \right) }^{ -4 }={ \left( \frac { 5 }{ 8 } \right) }^{ 4 }\)
Now, \({ \left( \frac { 5 }{ 8 } \right) }^{ -7 }\times { \left( \frac { 8 }{ 5 } \right) }^{ -4 }={ \left( \frac { 5 }{ 8 } \right) }^{ -7 }\times { \left( \frac { 5 }{ 8 } \right) }^{ 4 }\)
\(={ \left( \frac { 5 }{ 8 } \right) }^{ -7+4 }={ \left( \frac { 5 }{ 8 } \right) }^{ -3 }\)   [\(\because\) a^m \(\times\) aⁿ = a^m+n]
\(={ \left( \frac { 8 }{ 5 } \right) }^{ 3 }=\frac { 8\times 8\times 8 }{ 5\times 5\times 5 } =\frac { 512 }{ 125 } \)

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=\(\frac{1}{(-1)^{1}}\)     \( \left[ \because \quad { a }^{ -m }=\frac { 1 }{ { a }^{ m } } \right] \)
=\(\frac{1}{-1}=-1\)

We have,10^-5=\(\frac{1}{10^{5}}\)
Reciprocal of \(\frac{1}{10^{5}}\)=10⁵
∴ Multiplicative inverse of 10^-5 is 10⁵.    [\(10^{-5}\times 10^{5}=10^{0}=1\)]

We have ,2^-4=\(\frac{1}{2^{4}}\)
Reciprocal of=\(\frac{1}{2^{4}}\)=2^-4   [∵ reciprocal of \(\frac{1}{a}\)is a]
∴ Multiplicative inverse of 2^-4 is 2⁴.       [∵ \(2^{-4}\times2^{4}=2^{0}=1\)]