# Exercise 12.2 Exponents and Powers- NCERT Solutions Class 8

Exponents and Powers

Exercise 12.2

## Question 1

Express the following numbers in standard form.

(i) \(0.0000000000085\)

(ii) \(0.00000000000942\)

(iii) \(6020000000000000\)

(iv) \(0.00000000837\)

(v) \(31860000000\)

### Solution

**Video Solution**

(i) \(0.0000000000085\)

**What is known?**

Usual form

**What is unknown?**

Standard form

**Reasoning:**

How to use –\(a.b \times {10^n}\)

Where a is a whole number, b is a decimal number and n is an integer. Small numbers are expressed with negative exponent i.e. n is negative integer.

**Steps:**

To convert this small number in to standard from we need to move decimal to its right by \(12\) steps.

\(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | \(13\) | ||

\(0\) | \(\bf{.} \) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(8\) | \(5\) |

\[\begin{align}&0.0000000000085 \\&= \frac{{85}}{{10000000000000}}\\&= \frac{{85}}{{10000000000000}} \\&= \frac{{8.5 \times 10}}{{{{10}^{13}}}}\\&= 8.5 \times {10^{ - 13}} \times {10^1}\\&= 8.5 \times {10^{ - 12}}\end{align}\]

\(\therefore\) Standard form of given number is \(8.5 \times {10^{ - 12}}\)

(ii) \(0.00000000000942\)

**What is known?**

Usual form

**What is unknown?**

Standard form

**Reasoning:**

How to use – \(a.b \times {10^n}\)

Where a is a whole number, b is a decimal number and n is an integer. Small numbers are expressed with negative exponent i.e. n is negative integer.

**Steps:**

\(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | \(13\) | \(14\) | ||

\(0\) | \(\bf{.}\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(9\) | \(4\) | \(2\) |

\[\begin{align}&0.00000000000942 \\&= \frac{{942}}{{100000000000000}}\\&= \frac{{942}}{{100000000000000}}\\&= \frac{{9.42 \times {{10}^2}}}{{{{10}^{14}}}}\\&= 9.42 \times {10^{ - 14}} \times {10^2}\\&= 9.42 \times {10^{ - 12}}\end{align}\]

\(\therefore\) Standard form of given number is \(9.42 \times {10^{ - 12}}\)

(iii) \(6020000000000000\)

**What is known?**

Usual form

**What is unknown?**

Standard form

In this question a big number has to be converted to its standard form. In this case \(n\) is positive to represent this number as \(a \cdot b \times {10^{\rm{n}}}\) and the decimal will be moved to its left to represent this number in its standard form as bellow.

\(1\) |
\(2\) |
\(3\) |
\(4\) |
\(5\) |
\(6\) |
\(7\) |
\(8\) |
\(9\) |
\(10\) |
\(11\) |
\(12\) |
\(13\) |
\(14\) |
\(15\) |
\(16\) |

\(6\) |
\(0\) |
\(2\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |

\[6020000000000000 = 6.02 \times {10^{15}}\]

Decimal has moved by \(15\) steps so standard form of given number is \(6.2 \times {10^{15}}\)

(iv) \(0.00000000837\)

**What is known?**

Usual form

**What is unknown?**

Standard form

**Reasoning:**

How to use –\(a.b \times {10^n}\)

Where a is a whole number, b is a decimal number and \(n\) is an integer. Small numbers are expressed with negative exponent i.e. \(n\) is negative integer.

**Steps:**

\(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | ||

\(0\) | \(\bf{.}\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(8\) | \(3\) | \(7\) |

\[\begin{align}&0.000000000837 \\&= \frac{{837}}{{100000000000}}\\ &= \frac{{8.37 \times {{10}^2}}}{{{{10}^{11}}}}\\&= 8.37 \times {10^{ - 11}} \times {10^2}\\&= 8.37 \times {10^{ - 9}}\end{align}\]

\(\therefore\) Standard form of given number is \(8.37 \times {10^{ - 9}}\)

(v) \(31860000000\)

**What is known?**

Usual form

**What is unknown?**

Standard form

**Reasoning:**

In this question a big number has to be converted to its standard form. In this case *n* is positive to represent this number as \(a \cdot b \times {10^{\rm{n}}}\)and the decimal will be moved to its left to represent this number in its standard form as below.

**Steps:**

\(1\) |
\(2\) |
\(3\) |
\(4\) |
\(5\) |
\(6\) |
\(7\) |
\(8\) |
\(9\) |
\(10\) |
\(11\) |

\(3\) |
\(1\) |
\(8\) |
\(6\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
\(0\) |

\[31860000000 = 3.186 \times {10^{10}}\]

So, decimal has moved by \(10\) steps to its left

\(\therefore\) Standard form of given number is \(3.186 \times {10^{10}}\)

## Question 2

Express the following numbers in usual form.

(i) \(3.02 \times 10^{-6}\)

(ii) \(4.5 \times 10^{4}\)

(iii) \(\begin{align}\text{3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{-8}}\end{align}\)

(iv) \(1.0001 \times 10^{9}\)

(v) \(5.8 \times 10^{12}\)

(vi) \(3.61492 \times 10^{6}\)

### Solution

**Video Solution**

**What is known?**

Standard form

**What is unknown?**

Usual form

**Reasoning:**

As we know standard form of any small or larger number is \(a.b \times {10^n}\),Where \(a\) is a whole number, \(b\) is a decimal number and \(n\) is an integer

For small number \(n\) is negative and for large number \(n\) is positive. So, to convert a small number to its usual form we need to move decimal to its left by number of steps given as exponent values.

**Steps:**

(i) \(\,3.02 \times {10^{ - 6}}\)

\(\therefore\) Its usual form is

\[\begin{align}3.02 \times {10^{ - 6}} &= \frac{{3.02}}{{1000000}}\\&= 0.00000302\end{align}\]

(ii) \(\,4.5 \times {10^4}\)

Now to convert a big number to its usual form we need to move decimal to its right by number of steps given as its exponent.

\[4.5 \times {10^4}= 45000\]

\(\therefore\) Answer is \(45000\)

(iii) \(\,3 \times {10^{ - 8}} = {{ }}0.00000003\)

(iv) \(1.0001 \times {10^9} = 1000100000\)

(v) \(\,5.8 \times {10^{12}} = {{ }}5800000000000\)

(vi) \(\,3.61492 \times {10^6} = 3614920\)

## Question 3

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

(i) \(1\) micron is equal to \(\begin{align}\frac{1}{{100000}}\,\,{\rm{m}}\end{align}\)

(ii) Charge of an electron is \(0.000,000,000,000,000,000,16\) coulomb.

(iii) Size of a bacteria is \(0.0000005\, \rm{m}\)

(iv) Size of a plant cell is \(0.00001275\, \rm{m}\)

(v) Thickness of a thick paper is \(0.07 \,\rm{mm}\)

### Solution

**Video Solution**

**What is known?**

Usual form

**What is unknown?**

Standard form

**Reasoning:**

How to use –\(a.b \times {10^n}\)

Where a is a whole number, b is a decimal number and n is an integer. Small numbers are expressed with negative exponent i.e. \(n\) is negative integer.

**Steps:**

(i) \(1\) micron is equal to \(\begin{align}\frac{1}{{1000000}}\; \rm{m}\end{align}\)

\[\begin{align}\frac{1}{{1000000}} & =0.000001\\ &=1\times10^{-6}\end{align}\]

(ii) \(0.00000000000000000016\)

\[\begin{align}& 0.00000000000000000016\\ &=\frac{16}{100000000000000000000} \\& =\frac{1.6\times 10}{{{10}^{20}}} \\& =1.6\times {{10}^{-19}}\,\,\text{coulomb}\end{align}\]

(iii) \(0.0000005 \;\rm{m}\)

\[\begin{align}0.0000005\text{m}&=\frac{5}{10000000} \\& =\frac{5}{{{10}^{7}}} \\& =5\times {{10}^{-7}}\text{m}

\end{align}\]

(iv) \(0.00001275 \;\rm{m}\)

\[{\begin{align}0.00001275\,\text{m}~&=\frac{1275}{100000000} \\& =\frac{1.275\times {{10}^{3}}}{{{10}^{8}}} \\& =1.275\times {{10}^{-5}} \rm m\end{align}}\]

(v) \(0.07\,\text{mm}\)

\[\begin{align} 0.07\text{mm}&=\frac{7}{100} \\& =\frac{7}{{{10}^{2}}} \\& 7\times {{10}^{-2}}\text{mm} \end{align}\]

## Question 4

In a stack there are \(5\) books each of thickness \(20\, \rm{mm}\) and \(5\) paper sheets each of thickness \(0.016\,\rm{mm} \).

What is the total thickness of the stack?

### Solution

**Video Solution**

**What is known?**

\(5\) books of \(20\rm{mm}\) thickness each. \(5\) paper sheets of \(0.016\rm{mm}\) thickness each.

**What is unknown?**

Total thickness of the stack

**Reasoning:**

First find the total thickness of \(5\) books then find out total thickness of \(5\) paper sheets. Now find total thickness of stack by adding these two.

**Steps:**

Thickness of each book \(= 20 \,\rm{mm}\)

Number of books per stack \(= 5\)

Thickness of \(5\) books \(=5\times20=10\,\rm{mm}\)

Thickness of each sheet\(=0.016\,\rm{mm}\)

Thickness of 5 paper sheets

\(=5\times 0.016=0.080\,\,\text{mm}\)

Total thickness of stack

\[ \begin{align} &= 100 +0.080 \\ & =100.080\;\rm{mm}\\ & =100.080 \\ & =1.0008\times10^2\,\rm{mm} \end{align} \]

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