# Exercise 13.3 Exponents-and-Powers -NCERT Solutions Class 7

Go back to  'Exponents and Powers'

## Chapter 13 Ex.13.3 Question 1

Write the following numbers in the expanded forms:

$$279404,3006194,2806196,\\\qquad120719,20068$$

### Solution

Reasoning:

Expanded form of number means expressing a number using powers of $$10$$ as exponents.

(i) $$279404$$

\begin{align}&=\!\left[\begin{array} =2 \times 100000 + 7 \times 10000 +\\ 9 \times 1000 + 4 \times 100 +\\ 0 \times 10 + 4 \times 1\end{array} \right]\\&=\!\left[\begin{array}=2 \times {{10}^5} + 7 \times {{10}^4} + \\9 \times {{10}^3} + 4 \times {{10}^2} +\\ 0 \times {{10}^1} + 4 \times {{10}^0}\end{array} \right]\end{align}

(ii) $$3006194$$

\begin{align} &=\! \left[\begin{array} = 3 \times 1000000 + \\ 0 \times 10000 + 0 \times 10000 \\ + 6 \times 1000 + 1 \times 100 \\ + 9 \times 10 + 4 \times 1\end{array} \right]\\&= \left[\begin{array}= 3 \times {{10}^6}+0\times{{10}^5} +\\ 0\times {{10}^4} + 6\times {{10}^3} +\\1 \times{{10}^2} + 9 \times {{10}^1} +\\ 4 \times {{10}^0}\end{array} \right]\end{align}

(iii)  $$2806196$$

\begin{align}&=\left[\begin{array} = 2 \times 10000 + 0 \times 1000 + \\ 0 \times 100 + 6 \times10 + \\ 8 \times 1\end{array} \right]\\&=\left[\begin{array}= 2 \times {{10}^4} + 0 \times {{10}^3} +\\ 0 \times {{10}^2} + 6 \times {{10}^1} + \\ 8 \times {{10}^0}\end{array} \right]\end{align}

(iv)  $$120719$$

\begin{align}&=\! \left[\begin{array} = 1 \times 100000 + 2 \times 10000 + \\ 0 \times 1000 + 7 \times 100 + \\ 1 \times 10 + 9 \times 1\end{array} \right]\\&=\! \left[\begin{array} = 1 \times {10}^5 + 2 \times {10}^4 + \\ 0 \times {10}^3 + 7 \times {10}^2 + \\ 1 \times {10}^1 + 9 \times {10}^0\end{array} \right]\end{align}

(v) $$20068$$

\begin{align}&=\left[\begin{array} = 2 \times 10000 + 0 \times 1000 +\\ 0 \times 100 + 6 \times 10 + 8 \times 1\end{array} \right]\\&=\left[\begin{array}=2 \times {{10}^4} + 0 \times {{10}^3} + 0 \times\\ {{10}^2} + 6 \times {{10}^1} + 8 \times {{10}^0}\end{array}\right]\end{align}

## Chapter 13 Ex.13.3 Question 2

Find the number from each of the following expanded forms:

(a)

\left[ \begin{align} & 8\times {{10}^{4}}+6\times {{10}^{3}}+0\times \\ & {{10}^{2}}+4\times {{10}^{1}}+5\times {{10}^{0}} \\ \end{align} \right]

(b)

$$4 \! \times \! 10^5 \! + \! 5 \! \times \! 10^3 \! + \! 3 \! \times \! 10^2 \! + \! 2 \! \times \! 10^0$$

(c)

$$3 \times {{10}^4} + 7 \times {{10}^2} + 5 \times {{10}^0}$$

(d)

$$9 \times {{10}^5} + 2 \times {{10}^2} + 3 \times {{10}^1}$$

### Solution

Reasoning:

Expressing a number means we have to expand the following powers of $$10.$$

(a)

\left[ \begin{align} & 8\times {{10}^{4}}+6\times {{10}^{3}}+0\times \\ & {{10}^{2}}+4\times {{10}^{1}}+5\times {{10}^{0}} \\ \end{align} \right]

\begin{align} & =\left[ \begin{array} & 8\times 10000+6\times 1000+ \\ 0\times 100+4\times 10+5\times 1 \\ \end{array} \right] \\ & =80000+6000+0+40+5 \\ & =86045 \\ \end{align}

(b)

$$4 \! \times \! {{10}^5} \! + \! 5 \! \times \! {{10}^3} \! + \! 3 \! \times \! {{10}^2} \! + \! 2 \times \! {{10}^0}$$

\begin{align} & =\!\left[ \begin{array} & 4\times 100000+ 0\times 10000+\\ 5\times 1000+3\times 100+ \\0\times 10+2\times 1 \end{array}\right] \\ & =\!\left[ \begin{array} &400000+0+5000\\+300+0+2 \end{array}\right]\\ & = 405302 \end{align}

(c) $$3 \times {{10}^4} + 7 \times {{10}^2} + 5 \times {{10}^0}$$

\begin{align} & =\left[ \begin{array} & 3\times 10000+0\times 10000+ \\ 7\times 100+0\times 10+5\times 1 \\ \end{array} \right] \\ & =30000+0+700+0+5 \\ & =30705 \\ \end{align}

(d) $$9 \times {{10}^5} + 2 \times {{10}^2} + 3 \times {{10}^1}$$

\begin{align} & =\left[ \begin{array}& 9\times 10000+0\times 10000+ \\ 0\times 1000+ 2\times 100+ \\ 3\times 10+0\times 1 \end{array} \right] \\ & =900000+0+0+200+30+0 \\ & =900230 \\ \end{align}

## Chapter 13 Ex.13.3 Question 3

Express the following numbers in standard form:

(i)  $$5,00,00,000$$

(ii) $$70,00,000$$

(iii) $$3,18,65,00,000$$

(iv) $$3,90,878$$

(v)  $$39087.8$$

(vi) $$3908.78$$

### Solution

Reasoning:

Standard form of numbers is used in case of large numbers. Large numbers can be expressed using exponents of $$10$$.

(i)

$$5,00,00,000= 5\times{{10}^7}$$

(ii)

$$70,00,000 = 7 \times {{10}^6}$$

(iii)

$$3,18,65,00,000= 3.1865 \times {{10}^9}$$

(iv)

$$3,90,878= 3.90878 \times {{10}^5}$$

(v)

$$39087.8= 3.90878 \times {{10}^4}$$

(vi)

$$3908.78= 3.90878 \times {{10}^3}$$

## Chapter 13 Ex.13.3 Question 4

Express the number appearing in the following statements in standard form.

(a) The distance between Earth and Moon is $$384,000,000 \rm\, m$$

(b) Speed of light in vacuum is $$300,000,000 \rm \,m/s$$

(c) Diameter of the Earth is $$1,27,56,000\rm \,m$$

(d) Diameter of the Sun is $$1,400,000,000 \rm\,m$$

(e) In a galaxy there are on an average $$100,000,000,000$$ stars

(f) The universe is estimated to be about $$12,000,000,000$$ years old

(g) The distance of the Sun from the centre of the Milky Way Galaxy is estimated to be $$300,000,000,000,000,000,000 \rm\,m$$

(h) $$60,230,000,000,000,000,000,000$$ molecules are contained in a drop of water weighing $$1.8 \rm\,gm$$

(i) The earth has $$1,353,000,000$$ cubic $$\rm{km}$$ of sea water

(j) The population of India was about $$1,027,000,000$$ in March $$2001$$

### Solution

(a) The distance between Earth and Moon is $$384,000,000 \rm\, m$$.

Standard Form $$= 3.84\times10^{8} \rm\, m$$

(b) Speed of light in vacuum is $$300,000,000 \rm \,m/s$$.

Standard Form $$= 3\times10^{8} \rm \, m/s$$

(c) Diameter of the Earth is $$1,27,56,000\rm \,m$$.

Standard Form $$= 1.2756\times10^{7}\rm\, m$$

(d) Diameter of the Sun is $$1,400,000,000 \rm\,m$$.

Standard Form $$= 1.4\times10^{9} \rm\,m$$

(e) In a galaxy there are on an average $$100,000,000,000$$ stars.

Standard Form $$= 1\times10^{11}$$ stars

(f) The universe is estimated to be about $$12,000,000,000$$ years old.

Standard Form $$= 1.2\times10^{10}$$years old

(g) The distance of the Sun from the centre of the Milky Way Galaxy is estimated to be $$300,000,000,000,000,000,000 \rm\,m$$.

Standard Form $$= 3\times10^{20}\rm\, m$$

(h)  $$60,230,000,000,000,000,000,000$$ molecules are contained in a drop of water weighing $$1.8 \rm\,gm$$.

Standard Form $$= 6.023\times10^{22}$$ molecules

(i) The earth has $$1,353,000,000$$ cubic $$\rm{km}$$ of sea water.

Standard Form $$= 1.353\times10^{9} \rm cubic \,\rm\, km$$

(j) The population of India was about $$1,027,000,000$$ in March $$2001$$.

Standard Form $$= 1.027\times10^{9}$$