Introduction
In everyday life and science, we deal with very large numbers (distance from Earth to Sun) and very small numbers (size of a bacterium). Powers (also called exponents or indices) give us a compact way to write and compute with such numbers efficiently.
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Key Concepts
Exponent notation: am means 'a' multiplied by itself m times. Here 'a' is the base and 'm' is the exponent or power.
Example: 25 = 2 x 2 x 2 x 2 x 2 = 32.
- 1.Laws of Exponents (for non-zero integers a, b and integers m, n):
- 2.am x an = am+n — same base, add exponents
- 3.am / an = am-n — same base, subtract exponents
- 4.(am)n = am x n — power of a power
- 5.am x bm = (a x b)m — same exponent, multiply bases
- 6.am / bm = (a/b)m
- 7.a0 = 1 (a not equal to 0)
- 8.a-m = 1/am — negative exponent
Standard Form (Scientific Notation): Any number written as k x 10n where 1 ≤ k < 10 and n is an integer.
Example: 5,800,000 = 5.8 x 106.
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Worked Examples
Simplify 34 x 32.
Using law 1: 34+2 = 36 = 729.
Simplify (23)2.
Using law 3: 23 x 2 = 26 = 64.
Find the value of (2/3)-2.
(2/3)-2 = (3/2)2 = 9/4.
Express 0.000047 in standard form.
Move decimal 5 places right: 0.000047 = 4.7 x 10-5.
Simplify (52 x 54) / 53.
= 52+4-3 = 53 = 125.
Verify that 70 = 1 using the division law.
73 / 73 = 73-3 = 70. But any number divided by itself = 1. So 70 = 1.
Express the distance 149,600,000 km (Earth to Sun) in standard form.
149,600,000 = 1.496 x 108 km.
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Common mistakes
- Writing am x bn as (ab)m+n — this is WRONG. Law 4 only applies when exponents are equal.
- Confusing (-2)2 = 4 with -(22) = -4. Always apply the exponent before the sign unless brackets say otherwise.
- Forgetting that a0 = 1 for any non-zero a.
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Summary
Exponents provide a shorthand for repeated multiplication. The seven laws of exponents let us simplify complex expressions without multiplying everything out. Standard form is essential in science for handling very large or very small numbers neatly.