Exponents and Powers

12.1 Introduction

We learnt about the exponents in previous class. Any large number has converted in corresponds to the number of times the base is used as a factor known as exponent.

Power is an expression that represents repeated multiplication of the same factor.

Here we will learn more things about eh exponents and powers.

12.2 Powers with Negative Exponents

As the exponent decreases by1, the value becomes one-tenth of the previous value.

For any non-zero integer a, a^-m = 1/a^m, where m is a positive integer. a^−m is the multiplicative inverse of a^m.

10⁻² = 1/10² or 10² = 1/10²

12.3 Laws of Exponents

For any non-zero integer a, a^m x aⁿ = a^{m + n}, where m and n are natural numbers.

1. a^m/aⁿ = a^{m - n}

2. (a^m)ⁿ = a^mn

3. a^m x b^m

   = (ab)^m

4. a^m/b^m = (a/b)^m

5. a⁰ = 1

aⁿ = 1 only if n = 0. This will work for any a. For a = 1, 1¹ = 1² = 1³ = 1^{− 2} = ... = 1 or (1)ⁿ = 1 for infinitely many n.

For a = −1, (−1)⁰ = (−1)² = (−1)⁴ = (−1)⁻² = ... = 1 or (−1)^p = 1 for any even integer p.

(a/b)^-m = (b/a)^m

12.4 Use of Exponents to Express Small Numbers in Standard Form

When we have to add numbers in standard form, we convert the sign (positive or negative) of them into numbers with the same exponents.