**Points to Remembe of Exponents and Power**

- The numbers with negative exponents also obey the following laws:

(i) x^{m}* x^{n}= x^{m+n}

(ii) x^{m}÷ x^{n}= x^{m}^{–n}

(iii) x^{m}* b^{m}= (xb)^{m}

(iv) x^{0}= 1 - A number is said to be in the standard form, if it is expressed as the product of a number between 1 and 10 and the integral power of 10.
- Very small numbers can be expressed in standard form using negative exponents.

**We Know That**

When we write 5^{4}, it means 5 * 5 * 5 * 5, i.e. 5 is multiplied 4 times. So 5 is the base and 4 is the exponent. We read 5^{4} as “5 raised to the power 4’’.We also know the following laws of exponents

(i) x^{m} ⋅ x^{n} = x^{m}^{+n}

(ii) x^{m} ÷ x^{n} = x^{m}^{–n}

(iii) (x^{m})^{n} = x^{m*n}

(iv) x^{m} * y^{m} = (xy)^{m}

The value of any number raised to 0 is 1, i.e. a^{0} = 1.

We express very small or very large numbers in standard form (i.e scientific notation) for**Example:**

(ii) 3600000000000 = 36 * 10^{11} = 3.6 * 10^{12}

So, 3.6 * 10^{12} is the standard form.

**Power with Negative Exponents**

For a non-zero integer x, we have

or x^{–m} * x^{m} = 1

So x^{–m} is the reciprocal (or the multiplicative inverse) of x^{m} and vice versa.

For example: (i) Reciprocal of 8^{–7} = 8^{7} and

(ii) Reciprocal of 8^{7 }= 8^{–7}

**Solved Examples**

**Ques 1: Find the multiplicative inverse of the following.(i) 2 ^{–4}(ii) 10^{–5}(iii) 7^{–2}(iv) 5^{–3 }(v) 10^{–100}Solution: **(i) The multiplicative inverse of 2

^{–4}is 2

^{4}.

(ii) The multiplicative inverse of 10

^{–5}is 10

^{5}

(iii) The multiplicative inverse of 7

^{–2}is 7

^{2}.

(iv) The multiplicative inverse of 5

^{–3 }is 5

^{3}.

(v) The multiplicative inverse of 10

^{–100}is 10

^{100}.

**Ques 2: Find the value of Solution: **By using rule 2 and 3 –

**Ques 3: Solve the following: (-3) ^{2} × (5/3)^{3}**(-3)

Solution:

^{2}× (5/3)

^{3}

= (-3 × -3) × ( ( 5 × 5 × 5 ) / ( 3 × 3 × 3 ) )

= 9 × (125/27)

= (125/3)

**Ques 4: If x11 = y0 and x=2y, then y is equal to**

a. 1/2

b. 1

c. -1

d. -2

Solution:Option A. x

a. 1/2

b. 1

c. -1

d. -2

Solution:

^{11}= y

^{0}=> x

^{11}= 1 => x = 1. Given, x = 2y hence, y = x/2 =1/2

**Ques 5: By what number (4)**

Solution:4 Simplest way to to solve this would be:

^{-3}be multiplied so that the product become 1/16?Solution:

1/16 = 1/4

^{2}= (4)

^{-2 }

(4)

^{-3}× 4 = (4)

^{-2}

**Ques 6: What is the value of 6**

a. 18

b. 216

c. 729

d. 1296

^{3}?a. 18

b. 216

c. 729

d. 1296

**Solution:**Option b.

6^{3}

= 6 × 6 × 6

= 36 × 6

= 216**Ques 7****: What is the value of (-2) ^{-5}?**

**a. -0.03125**

b. 0.03125

c. 10

d. 32

b. 0.03125

c. 10

d. 32

**Solution:**Option A.

(-2)

^{-5}

= -0.03125