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

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Question

Express the following numbers in standard form.

(i) \(0.0000000000085\)

(ii) \(0.00000000000942\)

(iii) \(6020000000000000\)

(iv) \(0.00000000837\)

(v) \(31860000000\)

 Video Solution
Exponents And Powers
Ex 12.2 | Question 1

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

  
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