Exponents-Laws of Exponents

# Exponents-Laws of Exponents

## Exponents

We know that, 5 × 5 × 5 × 5 × 5 × 5 is written as 56 and is read as ‘5 raised to the power 6’. Here, 5 is called the base and 6 is called the exponent or power or index.
In general, if a is a literal number and it is multiplied by itself, m number of times, then  a × a × a × ... × a (m times) = am
This method of expressing product of a number by itself in short form is called the exponential notation or power notation.

Example 1: Express the following products in exponential form and evaluate.
a. 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
b. (–5) × (–5) × (–5) × (–5) × (–5)
Solution:
a. 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 = 78 = 5764801
b. (–5) × (–5) × (–5) × (–5) × (–5) = (–5)5 = –3125

## Laws of Exponents

If a and b are any two real numbers, and m and n are two any integers, then
Law 1: am × an = am + n
Law 2: am ÷ an = am – n, where a 0
Law 3: (am)n = am × n
Law 4: am × bm = (ab)m
Law 5: (a/b)m = am/bm
Law 6: (a/b)-m = (b/a)m , where a 0 and b ≠ 0

## Values with Zero Exponents

If a is any real number, then a0 = 1, where a 0.
For example, 120 = 1, (1245)0 = 1 and (235.121)0 = 1

## Values with Negative Exponents

If a is any real number, then a–n = 1/an and 1/a-n = an where a 0.
For example, 5-4 = 1/54 = 1/625 and 1/4-3 = 43 = 64

## Value with Fractional Exponents

If a is any real number, na = a1/n and nam = am/n where a 0.
For example, 38 = 81/3 and 573 = 73/5

Example 2: Evaluate the following using laws of exponents.
a. p × p × p × p × p × p × q × q × q × q × q × r
b. [(32)3]4
c. 20 × 50 × 2–3 × 5–5 × 53
d. (42)3 × 42 × 4–5

Solution:
a. p × p × p × p × p × p × q × q × q × q × q × r = p6q5r
b. [(32)3]4 = [32 × 3]4 = 36 × 4 = 324
c. 20 × 50 × 2–3 × 5–5 × 53 = 1 × 1 × 2–3 × 5–5 + 3 = 2–3 × 5–2 = 1/23 × 1/52 = 1/8 × 1/25 = 1/200
d. (42)3 × 42 × 4–5 = 42 × 3 × 42 × 4–5 = 4(6 + 2 – 5) = 43 = 64