1.1 Index Laws
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Learning Objectives
- Understand and apply the laws of indices for multiplication and division
- Apply the laws of indices for raising a power to another power
- Simplify expressions involving indices
- Solve equations involving indices
Key Concepts
Introduction to Indices
Indices (also called powers or exponents) are a mathematical shorthand for repeated multiplication. For example, 2³ means 2 × 2 × 2 = 8.
The number being multiplied (2 in this example) is called the base, and the number of times it's multiplied (3 in this example) is called the index or exponent.
Index laws provide rules for manipulating expressions with indices, making calculations more efficient.
The Multiplication Law
When multiplying two powers with the same base, we add the indices: x^a × x^b = x^(a+b)
This makes sense because x^a means x multiplied by itself a times, and x^b means x multiplied by itself b times. So together, that's x multiplied by itself (a+b) times.
Example 1
Simplify 2³ × 2⁴
Solution:
Using the multiplication law: 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128
Example 2
Simplify x² × x⁵
Solution:
Using the multiplication law: x² × x⁵ = x²⁺⁵ = x⁷
The Division Law
When dividing two powers with the same base, we subtract the indices: x^a ÷ x^b = x^(a-b)
This follows from the multiplication law. If we need to find how many times x^b goes into x^a, we're looking for the power that, when added to b, gives a.
Example 1
Simplify 5⁶ ÷ 5²
Solution:
Using the division law: 5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625
Example 2
Simplify y⁸ ÷ y³
Solution:
Using the division law: y⁸ ÷ y³ = y⁸⁻³ = y⁵
The Power of a Power Law
When raising a power to another power, we multiply the indices: (x^a)^b = x^(a×b)
This is because (x^a)^b means (x^a) multiplied by itself b times, which is x multiplied by itself a×b times.
Example 1
Simplify (3²)⁴
Solution:
Using the power of a power law: (3²)⁴ = 3²×⁴ = 3⁸ = 6,561
Example 2
Simplify (z³)⁵
Solution:
Using the power of a power law: (z³)⁵ = z³×⁵ = z¹⁵
The Power of a Product Law
When raising a product to a power, we raise each factor to that power: (x×y)^a = x^a × y^a
This distributes the exponent across each term in the product.
Example 1
Simplify (2×3)⁴
Solution:
Using the power of a product law: (2×3)⁴ = 2⁴ × 3⁴ = 16 × 81 = 1,296
Example 2
Simplify (ab)³
Solution:
Using the power of a product law: (ab)³ = a³b³
The Power of a Quotient Law
When raising a quotient to a power, we raise both the numerator and denominator to that power: (x/y)^a = x^a / y^a
This is similar to the power of a product law, but applies to division.
Example 1
Simplify (2/3)²
Solution:
Using the power of a quotient law: (2/3)² = 2² / 3² = 4/9
Example 2
Simplify (x/y)⁴
Solution:
Using the power of a quotient law: (x/y)⁴ = x⁴/y⁴
Test Your Knowledge
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