1.6 \(\times\) 10^-1

(-5)⁵

\(\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-5} =\frac{5^{-7}}{8^{-7}} \times \frac{8^{-5}}{5^{-5}}=\frac{5^{-7}}{5^{-5}} \times \frac{8^{-5}}{8^{-7}}=5^{(-7)-(-5)} \times 8^{(-5)-(-7)} \)
\(\\ =5^{-2} \times 8^{2}=\frac{8^{2}}{5^{2}}=\frac{64}{25}\)

\(\frac{1}{8} \times(3)^{-3}=\frac{1}{2^{3}} \times(3)^{-3}=2^{-3} \times 3^{-3}=(2 \times 3)^{-3}=6^{-3}=\frac{1}{6^{3}}\)

\(2^{-3}=\frac{1}{2^{3}}=\frac{1}{8}\)

4.05 \(\times\) 10⁶

10

3.72 x 10⁹ kg

∵ 1 hectare = 10000m²
∴ 5 hectares= 5 x 10000m²
= 50000m²= 5.0 x 10⁴m²

∵ 1km = 1000m
∴ 56km = 56000m = 5.6 x 10⁴ m

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Given, \(5^{m}\div 5^{-3}=5^{5}\) ⇒ \(\frac{5^{m}}{5^{-3}}=5^{5}\)     
⇒ 5^m-(-3) = 5⁵     \([\because \frac{a^{m}}{a^{n}}=a^{m-n}]\)
⇒ 5^m+3 = 5⁵
On comparing the powers of 5 of both sides, we get
m + 3 = 5 ⇒ m = 5 - 3 ⇒ m = 2
Hence, the value of m is 2.

\((\frac{1}{2})^{-5}\)=\(\frac{1^{-5}}{2^{-5}}\)    [∵ \((\frac{a}{b})^{-m}=\frac{a^{-m}}{b^{-m}}\)]
=\(\frac{2^{5}}{1^{5}}\)     [∵ \(a^{-m}=\frac{1}{a^{m}}\)]
= 2⁵ = 32    [(1)ⁿ=1]

We have, \((-2)^{-3}\times (-2)^{-4}=\frac{1}{(-2)^{3}}\times \frac{1}{(-2)^{4}}\)
[ஃ  a^m \(\times\) aⁿ = a^m+n, where m and n are integers]
\(=\frac{1}{(-2)^{3+4}}=\frac{1}{(-2)^{7}}=(-2)^{-7}\)
which is the required exponential form.

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