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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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