Points to Remembe of Exponents and Power

Points to Remembe of Exponents and Power

• The numbers with negative exponents also obey the following laws:
  (i) x^m * xⁿ = x^m+n
  (ii) x^m ÷ xⁿ = x^m–n
  (iii) x^m * b^m = (xb)^m
  (iv) x⁰ = 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 5⁴, 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 5⁴ as “5 raised to the power 4’’.We also know the following laws of exponents
(i) x^m ⋅ xⁿ = x^m+n
(ii) x^m ÷ xⁿ = 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. a⁰ = 1.
We express very small or very large numbers in standard form (i.e scientific notation) for
Example:

(ii) 3600000000000 = 36 * 10¹¹ = 3.6 * 10¹²
So, 3.6 * 10¹² 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 = 8⁷ and
(ii) Reciprocal of 8⁷ = 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 2⁴.
(ii) The multiplicative inverse of 10^–5 is 10⁵
(iii) The multiplicative inverse of 7^–2 is 7².
(iv) The multiplicative inverse of 5^–3 is 5³.
(v) The multiplicative inverse of 10^–100 is 10¹⁰⁰.

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

Ques 3: Solve the following: (-3)² × (5/3)³
Solution: (-3)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. x¹¹ = y⁰ => x¹¹ = 1 => x = 1. Given, x = 2y hence, y = x/2 =1/2

Ques 5: By what number (4)^-3 be multiplied so that the product become 1/16?
Solution: 4 Simplest way to to solve this would be:
1/16 = 1/4² = (4)^-2 
(4)^-3 × 4 = (4)^-2

Ques 6: What is the value of 6³ ?
a. 18
b. 216
c. 729
d. 1296
Solution: Option b.

6³
= 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
Solution: Option A.
(-2)^-5

= -0.03125