# Ex.12.2 Q1 Exponents and Powers - NCERT Maths Class 8

## Question

Express the following numbers in standard form.

(i) \(0.0000000000085\)

(ii) \(0.00000000000942\)

(iii) \(6020000000000000\)

(iv) \(0.00000000837\)

(v) \(31860000000\)

## Text 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 \(3.186 \times {10^{10}}\)