## Exponents And Powers

## Exponents And Powers Introduction

If a is any non-zero integer and n is a positive integer, then

x a (n times) is written as a”,

i.e., a” is the continued product of a multiplied by itself n times.

Here, ‘o’ is called the base, and V is called the ‘exponent’ or ‘index’.

The number a” is read as a raised to the power of or simply as ‘nth power of o’.

The notation a” is called the exponential or power notation.

We can write large numbers more conveniently using exponents.

**For example :**

10000 = 104; 243 = 35; 128 = 27, etc.

Now, we shall learn about negative exponents

## Powers With Negative Exponents

If a is any non-zero integer and m is a positive integer, then

⇒ \(a^{-m}=\frac{1}{a^m}\)

**Note:** a m is called the multiplicative inverse of am as a-m x am = 1

Am and a ~m are multiplicative inverses of each other.

Note 2: \(a^m=\frac{1}{a^{-m}}\)

**Question: What is 10 ~10 equal to?**

**Solution:** \(10^{-10}=\frac{1}{10^{10}}\)

**Question 1. Find the multiplicative inverse of the following:**

- 2-4
- 10-5
- 7-2
- 5-3
- 10-100

**Solution:**

The multiplicative inverse is \(2^{-4}=\left(\frac{1}{2^{-4}}\right) \text { is } 2^4\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

The multiplicative inverse of \(10^{-5}=\left(\frac{1}{10^{-5}}\right) \text { is } 10^5\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

The multiplicative inverse of \(7^{-2}=\left(\frac{1}{7^{-2}}\right) \text { is } 7^2\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

The multiplicative inverse \(5^{-3}=\left(\frac{1}{5^{-3}}\right) \text { is } 5^3\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

The multiplicative inverse\(10^{-100}=\left(\frac{1}{10^{-100}}\right) \text { is } 10^{100}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

**Question 2. Expand the following numbers using exponents :**

- 1025.63
- 1256.249

**Solution:**

1025.63

1 x 1000 + 0 x 100 + 2 x 10 + 5 x 1 + 6 x\(\frac{1}{10}+3 \times \frac{1}{100}\)

= 1 x 103 + 0 x 102 + 2 x 101 + 5 x 10° + 6 x 10-1 + 3 x 10-2

1256.249

= 1 x 1000 + 2 x 100 + 5 x 10

+ 6x 1+ 2 x \( \frac{1}{10}+4 \times \frac{1}{100}+9 \times \frac{1}{1000}\)

= 1 X 103 + 2 X 102 + 5 X 101 + 6 x 10° + 2 x 10-1 + 4 x 10-2 + 9 x lO-3

## Laws Of Exponents

If a, b are non-zero integers and m, n are any integers, then

- am x an = am+ n
- \(\frac{a^m}{a^n}=a^{m-n}\)
- (am)n = amn
- am x bm = (ab)m
- \(\frac{a^m}{b^m}=\left(\frac{a}{b}\right)^m\)
- a° = 1
- \(\left(\frac{a^{-m}}{b^{-n}}\right)=\frac{b^n}{a^m}\)
- \(\left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^m\)

## Remember

an” = 1 = n = 0

1n = 1 where n is any integer.

(- 1)n = 1 where n is any even integer.

(-1)n =-1 where n is any odd integer.

Q. Simplify and write in exponential form:

(-2)-3X (-2)-4

p3 x p -I0

32 x 3-5 x 36

Solution:

(-2)~3 x (-2)-4 = (- 2)(“3) + (-4>

am x an = am +n

= (-2)-7 = {(-1) x 2} -7

⇒ \(\frac{1}{\{(-1) \times 2\}^7}= \frac{1}{(-1)^7 \times(2)^7}\)

(ab)m = am bm

⇒ \(\frac{1}{(-1) \times 2^7}=-\frac{1}{2^7}\)

⇒ \(\mid(-1)^{\text {odd integer }}=-1\)

⇒ \(p^3 \times p^{-10}=p^{3+(-10)}=p^{-7}=\frac{1}{p^7}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

32 x 3-5 x 36 = 32 + (-5) + 6 = 33

## Exponents And Powers Exercise 10.1

**Question 1. Evaluate:**

- 3-2
- (-4)-2
- \(\left(\frac{1}{2}\right)^{-5}\)

**Solution:**

⇒ \(3^{-2}=\frac{1}{3^2}=\frac{1}{9}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \((-4)^{-2}=\frac{1}{(-4)^2}=\frac{1}{16}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(\left(\frac{1}{2}\right)^{-5}=\left(\frac{2}{1}\right)^5\)

⇒ \( a^{-m}=\frac{1}{a^m}\)

⇒ \(\frac{2^5}{1^5}\)

⇒ \(\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\)

⇒ \(\frac{32}{1}=32\)

1n = 1 where n is an integer

**Question 2. Simplify and express the result in power notation with a positive exponent**

- \((-4)^5 \div(-4)^8\)
- \(\left(\frac{1}{2^3}\right)^2\)
- \((-3)^4 \times\left(\frac{5}{3}\right)^4\)
- (3-7 – 3-10) x 3-5
- 2-3 x (-7) -3

**Solution:**

1. (-4)6 ÷ (-4)8

⇒ \(\frac{(-4)^5}{(-4)^8}\)

⇒ \((-4)^{5-8}\)

⇒ \((-4)^{-3}\frac{a^m}{a^n}=a^{m-n}\)

⇒ \(=\frac{1}{(-4)^3} a^{-m}=\frac{1}{a^m}\)

which is the required form.

2. \(\left(\frac{1}{2^3}\right)^2 =\frac{1^2}{\left(2^3\right)^2}\)

⇒ \(\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\)

⇒ \(\frac{1}{2^{3 \times 2}} \)

⇒ \( \mid\left(a^m\right)^n=a^m\)

⇒ \(\frac{1}{2^6}\)

which is the required form

3. \((-3)^4 \times\left(\frac{5}{3}\right)^4=\{(-1) \times 3\}^4 \times \frac{5^4}{3^4} \)

⇒ \(\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\)

⇒ \((-1)^4(3)^4 \times \frac{5^4}{3^4}\)

I (ab)m = am bm

= (- 1)4 x 54

= 1 x 54

(-1) even integer = 1

= 54

which is the required form.

4. (3-7 / 3-10) x 3-5

⇒ \(=\frac{3^{-7}}{3^{-10}} \times \frac{1}{3^5}\)

⇒ \( a^{-m}=\frac{1}{a^m}\)

⇒ \(3^{-7-(-10)} \times \frac{1}{3^5}\)

⇒ \(\frac{a^m}{a^n}=a^{m-n}\)

⇒ \(3^3 \times \frac{1}{3^5} \)

⇒ \(\frac{a^m}{a^n}=a^{m-n}\)

⇒ \(3^{3-5} \)

⇒ \(3^{-2} \)

⇒ \(a^{-m}=\frac{1}{a^m}\)

which is the required form.

5. \(2^{-3} \times(-7)^{-3}\)

⇒ \(\frac{1}{2^3} \times \frac{1}{(-7)^3}\)

⇒ \(\frac{1}{[2 \times(-7)]^3}=\frac{1}{a^m}\)

(ab)m = amb

⇒ \(\frac{1}{(-14)^3}\)

which is the required form.

**Question 3. Find the value of:**

(3° + 4 -1) x 22

(2-1 x 4-1)/2 -2

⇒ \(\left(\frac{1}{2}\right)^{-2}+\left(\frac{1}{3}\right)^{-2}+\left(\frac{1}{4}\right)^{-2}\)

⇒ \(\left(3^{-1}+4^{-1}+5^{-1}\right)^0\)

⇒ \(\left\{\left(\frac{-2}{3}\right)^{-2}\right\}^2.\)

**Solution:**

1. (3° + 4-1) x 22

⇒ \(\left(1+\frac{1}{4}\right) \times 4\)

⇒ \(a^{-m}=\frac{1}{a^m}, \quad a^0=1\)

⇒ \(\frac{5}{4} \times 4=5\)

(2-1 x 4-1) + 2″2

= {2 -l x (22)-1} + 2 -2

= {2-1 x 22x(-1>} -f2-2

(am)n = amn

= (2-’X2-2)T2-2 = 2(-1)+(-2) 2-2

am x a” = am+”

= 2-3 4- 2″2

⇒ \(\frac{2^{-3}}{2^{-2}}=2^{-3-(-2)} \)

⇒ \(\frac{a^m}{a^n}=a^{m-n}\)

⇒ \(2^{-1}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(\frac{1}{2}\)

3.\(\left(\frac{1}{2}\right)^{-2}+\left(\frac{1}{3}\right)^{-2}+\left(\frac{1}{4}\right)^{-2}\)

⇒ \(\frac{1^{-2}}{2^{-2}}+\frac{1^{-2}}{3^{-2}}+\frac{1^{-2}}{4^{-2}}\)

⇒ \(\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\)

⇒ \(\frac{2^2}{1^2}+\frac{3^2}{1^2}+\frac{4^2}{1^2}\)

⇒ \(\frac{4}{1}+\frac{9}{1}+\frac{16}{1}\)

= 4 + 9 + 16 = 29

4. [3-1 + 4-1 + 5-1]0

⇒ \({\left[\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right]^0 }\)

⇒ \(\left(\frac{20+15+12}{60}\right)^0\)

⇒ \(\quad \text { | LCM }(3,4,5)=60\)

⇒ \(\left(\frac{47}{60}\right)^0=1\)

\(a^0=1\)Aliter

(3-1 + 4-1 +5-1)° = 1

a°= 1

5. \(\left\{\left(\frac{-2}{3}\right)^{-2}\right\}^2\)

⇒ \(\left(\frac{-2}{3}\right)^{(-2) \times 2}\)

⇒ \(\mid\left(a^m\right)^n=a^{m n}\)

⇒ \(\left(\frac{-2}{3}\right)^{-4}\)

⇒ \(\left(\frac{3}{-2}\right)^4 \)

⇒ \( \left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^m\)

⇒ \(\frac{3^4}{(-2)^4}=\frac{3^4}{(-1 \times 2)^4}\)

⇒ \(\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\)

⇒ \(\frac{3^4}{(-1)^4(2)^4}\)

⇒ \((a b)^m=a^m b^m\)

⇒ \(\frac{3^4}{1 \times 2^4} \)

⇒ \((-1)^{\text {even integer }}=1\)

⇒ \(\frac{3^4}{2^4}\)

⇒ \(\frac{3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 2}=\frac{81}{16}\)

**Question 4. Evaluate:**

- \(\frac{8^{-1} \times 5^3}{2^{-4}}\)
- \(\left(5^{-1} \times 2^{-I}\right) \times 6^{-1}\)

**Solution:**

1. \(\frac{8^{-1} \times 5^3}{2^{-4}}\)

⇒ \(\frac{2^4 \times 5^3}{8^1}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(\frac{16 \times 125}{8}=250\)

2. \(\left(5^{-1} \times 2^{-1}\right) \times 6^{-1}=\left(\frac{1}{5} \times \frac{1}{2}\right) \times \frac{1}{6}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(\frac{1}{10} \times \frac{1}{6}=\frac{1}{60}\)

**Question 5. Find the value of m for which 5m + 5-3 = 55**

**Solution:**

5m + 5-3 = 55

⇒ \(\frac{5^m}{5^{-3}} =5^5\)

⇒ \(5^{m-(-3)} =5^5\)

⇒ \(\frac{a^m}{a^n}=a^{m-n}\)

⇒ \(5^{m+3} =5^5\)

bases are equal exponents are equal

m + 3 = 5

m = 5-3

m = 2

**Question 6. Evaluate:**

- \(\left\{\left(\frac{1}{3}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}\)
- \(\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-4}.\)

**Solution:**

⇒ \(\left\{\left(\frac{1}{3}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}\)

⇒ \(\left(\frac{1^{-1}}{3^{-1}}-\frac{1^{-1}}{4^{-1}}\right)^{-1}\)

⇒ \(\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\)

⇒ \(\left\{\frac{\frac{1}{1^1}}{\frac{1}{3^1}}-\frac{\frac{1}{1^1}}{\frac{1}{4^1}}\right\}^{-1}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(\left(\frac{3^1}{1^1}-\frac{4^1}{1^1}\right)^{-1}\)

⇒ \(a^1=a\)

⇒ \(\left(\frac{3}{1}-\frac{4}{1}\right)^{-1}\)

⇒ \( (3-4)^{-1}\)

⇒ \((-1)^{-1}=\frac{1}{(-1)^1}\)

⇒ \(| a^{-m}=\frac{1}{a^m}\)

⇒ \(\frac{1}{(-1)}\)

\((-1)^{\text {odd integer }}=-1\)= -1

⇒ \(\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-4}=\frac{5^{-7}}{8^{-7}} \times \frac{8^{-4}}{5^{-4}}\)

⇒ \(\frac{5^{-7}}{5^{-4}} \times \frac{8^{-4}}{8^{-7}}\)

⇒ \(=5^{(-7)-(-4)} \times 8^m=\frac{a^m}{b^m}\)

⇒ \(\frac{a^m}{a^n}=a^{m-n}\)

⇒ \(5^{-7+4} \times 8^{-4}+7 \)

⇒ \(\frac{1}{5^3} \times 8^3=\frac{8^3}{5^3}\)

⇒ \(\frac{8 \times 8 \times 8}{5 \times 5 \times 5}=\frac{512}{125}\)

**Question 7. Simplify:**

- \(\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}} \quad(t \neq 0)\)
- \(\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}\)

**Solution:**

1. \(\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}}\)

⇒ \(\frac{25 \times 5^3}{10} \times \frac{t^8}{t^4}\)

⇒ \(\frac{625}{2} \times \frac{t^8}{t^4}\)

⇒ \(\frac{625 t^4}{2}\)

⇒ \(\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}\)

⇒ \(\frac{3^{-5} \times(2 \times 5)^{-5} \times(5 \times 5 \times 5)}{5^{-7} \times(2 \times 3)^{-5}}\)

⇒ \(\frac{3^{-5} \times 2^{-5} \times 5^{-5} \times 5^3}{5^{-7} \times 2^{-5} \times 3^{-5}} \)

⇒ \(\quad(a b)^m=a^m b^m\)

⇒ \(\frac{5^{-5} \times 5^3}{5^{-7}}=\frac{5^{(-5)+3}}{5^{-7}}\)

⇒ \(\mid a^m \times a^n=a^{m+n}\)

⇒ \(\frac{5^{-2}}{5^{-7}}=5^{(-2)-(-7)}\)

⇒ \(5^{-2+7}=5^5\)

## Use Of Exponents To Express Small Numbers In Standard Form

A number is said to be in standard form if expressed in the form K x 10″ where 1 < K < 10 and n is an integer. A number written in standard form is said to be expressed in scientific notation.

Tiny numbers can be expressed in standard form using negative exponents.

1. To express a large number, we move the decimal point to the left such that only one digit is left to the left side of the decimal point and multiply the resulting number by 10n where n is the number of places to which the decimal point has been moved to the left.

**For example :** 270,000,000,000 = 2.7 x 1011

(Decimalpointhas have been moved to the left for11 places)

2. To express a number (< 1), we move the decimal point to the right such that only one digit is left to the left side of the decimal point and multiply the resulting number by 10~’1 where n is the number of places to which the decimal point has been shifted to the right.

**For example :** 0.000 0009 = 9 x 10^{-7}

(Decimalpointhas have been shifted to the right for 7 places.)

**Question 1. Identify huge and very small numbers from the above facts and write them in the following table:**

**Solution:**

**Question 2. Write the following numbers in standard form:**

- 0.000000564
- 0.0000021
- 21600000
- 15240000

**Solution:**

**1. 0.000000564**

= 5.64 x 10^{-7}

Moving decimal 7 places to the right

**2. 0.0000021**

0.0000021 = 2.1 x 10^{-6}

Moving decimal 6 places to the right

**3. 21600000**

21600000 = 2.16 x 10^{7}

Moving decimal 7 places to the left

**4. 15240000**

15240000 = 1.524 x 10^{7}

Moving decimal 7 places to the left

**Question 2. Write all the facts given in the standard form.**

**Solution:**

(1) The distance from the Earth to the Sun is 1.496 x 10^{11} m

149, 600,000,000 = 1.496 x 10^{11}.

Moving the decimal 11 places to the left

(2) The speed of light is 3 x 10^{8} m/sec.

300, 000, 000 = 3 x10^{8}.

Moving decimal 8 places to the left

The thickness of the Class VII Mathematics book is 2 x 10^{1} mm.

20 = 2 x 10 = 2 x 10^{1}

The average diameter of a Red Blood Cell is 7 x 10 6 mm.

0.00 0007 = 7 x 10^{-6}

Moving decimal 6 places to the right

The thickness of human hair is in the range of 5 x 10 -3 cm to 1 x I0″2cm

⇒ \(0.005=\frac{5}{1000}=\frac{5}{10^3}=5 \times 10^{-3}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(0.01=\frac{1}{100}=\frac{1}{10^2}=1 \times 10^{-2}\)

The distance of the moon from the Earth is 3.84467 x 10^{8}m

384,467,000 = 3.84467 x 10^{8}

I Moved the decimal 8 places to the left

(7) The size of a plant cell is 1.275 x 10^{-5}m

0.00001275 = 1.275 x 10^{5}

Moving decimal 5 places to the right

The average radius of the Sun is 6.95 x10^{5}km

695000 = 695 x 1000 = 695 x 10^{3} = 6.95 x 10^{5}

(9) Mass of fuel in a space shuttle

solid rocket booster is 5.036 x 10^{5} kg

503600

= 5036 x 100 = 5036 x 10^{2}

= 5.036 x 10^{3} x 10^{2}

= 5.036 x 10^{3+2}

a^{m} x a^{n} = a^{m+n}

= 5.036 x 10^{5}

(10) Thickness of a piece of paper is 1.6 x10^{-3} cm

0.0016 = 1.6 x 10^{-3}

Moving decimal 3 places to the right

The diameter of wire on a computer chip is 3 x 10^{-6} m

is 3 x 10^{-6}

0.000003 = 3 x 01^{3}

Moving decimal 6 places to the right

(12) The height of Mount Everest is 8.848 x 10^{3} m.

8848 = 8.848 x 1000 = 8.848 x 10^{3}

## Exponents And Powers Exercise 10.2

**Question 1. Express the following numbers in standard form:**

- 0.0000000000085
- 0.00000000000942
- 6020000000000000
- 0.00000000837
- 31860000000.

**Solution:**

**1. 0.0000000000085**

0.0000000000085 = 8.5 x 10^{-12}

Moving the decimal 12 places to the right

**2. 0.00000000000942**

0.00000000000942 = 9.42 x 10^{-15}

Moving the decimal 12 places to the right

**3. 6020000000000000**

6020000000000000 = 6.02 x 10^{15}

Moving the decimal 15 places to the left

**4. 0.00000000837**

0.00000000837 = 8.37 x 10^{-9}

Moving decimal 9 places to the right

**5. 31860000000**

31860000000 = 3.186 x 10^{10}

Moving decimal 10 places to the left

**Question 2. Express the following numbers in the usual form:**

- 3.02 x 10
^{-6} - 4.5 x 10
^{4} - 3 x 10
^{-8} - 1.0001 x 10
^{9} - 5.8 x 10
^{12} - 3.61492 x 10
^{6}

**Solution:**

⇒ \(3 \times 10^{-8}=\frac{3}{10^8}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(=\frac{3.02}{1000000}\)

⇒ 0.00000302

**2. 4.5 x 10 ^{4} = 4.5 x 10000 = 45000**

**3. ** \(3.02 \times 10^{-6}=\frac{3.02}{10^6}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(\frac{3}{100000000}\)

=0.00000003

**4. 1.0001 X 10 ^{9} **

= 1.0001 X 1000,000,000

= 1,000,100,000

**5. 5.8 X 10 ^{12} **

= 5.8 X 1,000,000,000,000

= 5,800,000,000,000

**6. 3.61492 x 10 ^{6} **

= 3.61492 x 1,000,000

= 3,614,920

**Question 3. Express the number appearing in the following statements in standard form:**

- 1 micron is equal to \(\frac{1}{1000000} m\)
- The charge of an electron is 0. 000, 000,000,000,000,000,1 6 coulomb.
- The size of bacteria is 0. 0000005 m
- The size of a plant cell is 0. 00001 275 m
- The thickness of thick paper is 0.07 mm.

**Solution:**

**1. ** \(\frac{1}{1000000} \mathrm{~m}= \frac{1}{10^6}\)

⇒ \(1 \times 10^{-6} \mathrm{~m}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

which is the required standard form.

**2. 0.000,000,000,000,000,000,16 coulomb**

⇒ \(\frac{16}{100,000,000,000,000,000,000} \text { coulomb }\)

⇒ \(frac{16}{10^{20}} \text { coulomb }\)

⇒ \(\frac{1.6 \times 10}{10^{20}} \text { coulomb }\)

⇒ \(\frac{1.6 \times 10^1}{10^{20}} \text { coulomb } \quad \mid a^1=a\)

⇒ \(1.6 \times 10^{1-20} \text { coulomb } \frac{a^m}{a^n}=a^{m-n}\)

⇒ \(1.6 \times 10^{-19} \text { coulomb }\)

which is the required standard form.

**3. 0.0000005 m**

⇒ \(\frac{5}{10000000} \mathrm{~m} \)

⇒ \(\frac{5}{10^7} \mathrm{~m}\)

⇒ \(5 \times 10^{-7} \mathrm{~m}\)

which is the required standard form.

4. \(0.00001275 \mathrm{~m} =\frac{1275}{100,000,000} \mathrm{~m}\)

⇒ \(\frac{1275}{10^8} \mathrm{~m}\)

⇒ \(\frac{1275}{10^3 \times 10^5} \mathrm{~m}\)

| am x a” = am + n

⇒ \(\frac{1275}{10^3} \times 10^{-5} \mathrm{~m}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

⇒ \(1.275 \times 10^{-5} \mathrm{~m}\)

which is the required standard form.

**5. \( 0.07 \mathrm{~mm} =\frac{7}{100}\)**

⇒ \(\frac{7}{10^2}=7 \times 10^{-2} \mathrm{~mm}\)

⇒ \(a^{-m}=\frac{1}{a^m}\)

which is the required standard form

**Question 4. In a stack, there are 5 books each of thickness 20 mm, and 5 paper sheets each of thickness 0. 016mm. What is the total thickness of the stack?**

**Solution:**

Total thickness of books = 5 x 20 mm = 100 mm

Total thickness of paper sheets = 5 x 0.016 mm = 0.080 mm

Total thickness of the stack

= Total thickness of books + Total thickness of paper sheets

= 100 mm + 0.080 mm

= (100 + 0.080) mm

= 100.080 mm

= 1.0008 x 10^{2} mm.

Moving decimal 2 places to the left

Hence, the total thickness of the stack is 1.0008 x IQ2 mm.

## Exponents And Powers Multiple-Choice Question And Solutions

**Question 1. a ^{m} x a^{n} is equal to**

- a
^{m+n} - a
^{m-n} - a
^{mn} - a
^{n-m}

**Solution:** 1. a^{m+n}

**Question 2. a ^{m} ÷ a^{m} is equal to**

- a
^{m+n} - a
^{m-n} - a
^{mn} - a
^{n-m}

**Solution:** 1. a^{m+n}

**Question 3. (a ^{m})is equal to**

- a
^{m+n} - a
^{m-n} - a
^{mn} - a
^{n-m}

**Solution:** 3. a^{mn}

**Question 4. Am x is equal to**

- (ab)
^{m} - (ab)
^{-m} - a
^{m}b - ab
^{m}

**Solution:** 1. (ab)^{m}

**Question 5. a0 is equal to**

- 0
- 1
- -1
- a

**Solution:** 2. 1

**Question 6. \(\frac{a^m}{b^m}\) is equal to**

- \(\left(\frac{a}{b}\right)^m\)
- \(\left(\frac{b}{a}\right)^m\)
- \(\frac{a^m}{b}\)
- \(\frac{a}{b^m}\)

**Solution:** 1. \(\left(\frac{a}{b}\right)^m\)

**Question 7. 2 x 2 x 2 x 2 x 2 is equal to**

- 24
- 23
- 22
- 25

**Solution:** 4. 25

**Question 8. In 102, the exponents**

- l
- 2
- 10
- 1

**Solution:** 2. 2

**Question 9. In 102, the base is**

- 1
- 0
- 10
- 100

**Solution:** 3. \(\frac{1}{10}\)

**Question 10. 10 ^{-1} is equal to**

- 10
- -1
- \(\frac{1}{10}\)
- \(-\frac{1}{10} \text {. }\)

**Solution:** 3. \(\frac{1}{10}\)

**Question 11. The multiplicative inverse of 2 ^{-3} is**

- 2
- 3
- -3
- 2
^{3}

**Solution:** 4. 2^{3}

**Question 12. The multiplicative inverse of 10 ^{5} is**

- 5
- 10
- 10
^{-5} - 10
^{5}

**Solution:** 3. 10^{-5}

**Question 13. The multiplicative inverse of \( \frac{1}{2^2}\)**

- 2
^{-2} - 2
^{2} - 2
- 1

**Solution:** 2. 22

**Question 14. The multiplicative inverse of 10 “10 is**

- 10
- \(\frac{1}{10}\)
- 10
^{-10} - 10
^{10}

**Solution:** 4. 10^{10}

**Question 15. The multiplicative inverse of a ^{m} is**

- a
- m
- a
^{m} - a
^{-m}

**Solution:** 5. a^{-m}

**Question 16. 5 ^{3} x 5^{-1} is equal to**

- 5
- 5
^{3} - 5
^{-1} - 5
^{2}

**Solution:** 4. 5^{2}

**Question 17. (-2) ^{5} x (- 2)^{6} is equal to**

- 2
- -2
- -5
- 6

**Solution:** 2. -2

**Question 18. 3 ^{2} x 3^{-4} x 3^{5} is equal to**

- 3
- 3
^{2} - 3
^{3} - 3
^{5}

**Solution:** 3. 3^{3}

**19. (- 2) 2 is equal to**

- \(\frac{1}{4}\)
- \(\frac{1}{2}\)
- \(-\frac{1}{2}\)
- \(-\frac{1}{4}\)

**Solution:** 1. \(\frac{1}{4}\)

**Question 20. \(\left(\frac{1}{2}\right)^{-4}\) is equal to**

- 2
- 2
^{4} - 1
- 2
^{-4}

**Solution:** 2. 2^{4}

**Question 21. (2 ^{0} + 4^{-1}) x 22 is equal to**

- 2
- 3
- 4
- 5

**Solution:** 4. 5

**Question 22. (2 ^{-1} + 3^{-1} + 5^{-1})^{0} is equal to**

- 2
- 3
- 5
- 1

**Solution:** 4. 1

**Question 23. 3 ^{m}÷ 3^{-3} = 3^{5} ⇒ m is equal to**

- 1
- 2
- 3
- 4

**Solution:** 2. 2

**Question 24. (-2) ^{m+1} x (-2)^{4} = (- 2)^{6 }⇒ m =**

- 0
- 1
- -1
- none of these

**Solution:** 2. 1

**Question 25. (-1) ^{60} is equal to**

- -1
- 1
- 50
- -50

**Solution:** 2. 1

**Question 26. (-1) ^{51} is equal to**

- -1
- 1
- 51
- -51

**Solution:** 1. -1

**Question 27. 149600000000 is equal to**

- 1.496 x 10
^{11} - 1.496 x lO
^{10} - 1.496 x 10
^{12} - 1.496 x 10
^{5}

**Solution:** 1. 1.496 x 10^{11}

**Question 28. 300000000 is equal to**

- 3 x 10
^{8} - 3 x 10
^{7} - 3 x 10
^{6} - 3 x 10
^{9}

**Solution:** 1. 3 x 10^{8}

**Question 29. 0.000007 is equal to**

- 7 x 10
^{-6} - 7 x 10
^{-6} - 7 x 10
^{-4} - 7 x 10
^{-3}

**Solution:** 1. 7 x 10^{-6}

**Question 30. 384467000 is equal to**

- 3.84467 x 1o
^{8} - 3.84467 x 10
^{3} - 3.84467 x 10
^{7} - 3.84467 x 10
^{6}

**Solution:** 1. 3.84467 x 10^{8}

**Question 31. 0.00001275 is equal to**

- 1.275 x 10
^{-6} - 1.275 x 10
^{-3} - 1.275 x 10
^{4} - 1.275 x 10
^{3}

**Solution:** 1. 1.275 x 10^{-6}

**Question 32. 695000 is equal to**

- 6.95 x 10
^{5} - 6.95 x 10
^{3} - 6.95 x 10
^{6} - 6.95 x 10
^{4}

**Solution:** 1. 6.95 x 10^{5}

**Question 33. 503600 is equal to**

- 5.036 x 10
^{5} - 5.036 x 10
^{6} - 5.036 x 10
^{4} - 5.036 x 10
^{7}

**Solution:** 1. 5.036 x 10^{5}

**Question 34. 0.0016is equal to**

- 1.6 x 10
^{-3} - 1.6 x 10
^{-2} - 1.6 x 10
^{-4} - 1.6 x lO
^{-5}

**Solution:** 1. 1.6 x 10^{-3}

**Question 35. 0.000003 is equal to**

- 3 x 10
^{-6} - 3 x 10
^{6} - 3 x 10
^{5} - 3 x 10
^{-5}

**Solution:** 1. 3 x 10^{-6}

**Question 36. 8848 is equal to**

- 8.848 x 10
^{3} - 8.848 x 10
^{2} - 8.848 x 10
- 8.848 x 10
^{4}

**Solution:** 1. 8.848 x 10^{3}

**Question 37. 1.5 x 10 ^{11} is equal to**

- 150000000000
- 15000000000
- 1500000000
- 1500000000000

**Solution:** 1. 150000000000

**Question 38. 2.1 x 10 ^{-6} is equal to**

- 0.0000021
- 0.000021
- 0.00021
- 0.0021.

**Solution:** 1. 0.0000021

**Question 39. 2.5 x 10 ^{4} is equal to**

- 25
- 250
- 2500
- 25000

**Solution:** 4. 25000

**Question 40. 0.07 x 1O ^{10}is equal to**

- 700000000
- 7000000
- 7000
- 7

**Solution:** 1. 700000000

## Exponents And Powers True-False

**Write whether the following statements are True or False:**

1. The value of \(\left\{(-1)^{-1}\right\}^{-1}\) is 1: **False**

2. The reciprocal of \(\left(\frac{4}{3}\right)^0\) is 1: **True**

3. The standard form of \(\frac{1}{1000000}\) is 1.0 x 10 -6: **True**

4. If 6m + 6 “3 = 66, then the value of m is 3: **True**

5. 2345.6 = 2 x 1000 + 3 x 100 + 4 x 10 + 5 x 1 + 6 x 10 – 1: **True**

## Exponents And Powers Fill in the Blanks

1. (1000)° = 1

2. The standard form of 1,234,500,000,000 is: **1.2345 x 1012**

3. The multiplicative inverse of(-3) ~2 is: **(-3)2**

4. (- 9)4 -5- (- 9)10 is equal to** (-9)-6**

5. The value of (2 -1 + 3 -1 + 4 -1)° is : **1**

6. Write 1.0002 x 109 in the usual form: **1000200000**

7. Write the reciprocal of 10 _1: **10**

8. Find the value of*if* “3 = (100)1-4 + (100)°: **100**

9. By what number should (-8) -1 be divided, so that the quotient may be equal to (-8) -1: 1

10. If = \(\frac{5^m \times 5^2 \times 5^{-3}}{5^{-5}}=5^4\) then find the value of m:** 0**