CIE IGCSE Mathematics (0580) Indices II Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Indices II Study Notes
LEARNING OBJECTIVE
- Understanding and Using Indices
Key Concepts:
- Indices
- Rules of Indices
Understanding and Using Indices
Understanding and Using Indices
What Are Indices?
Indices (also known as powers or exponents) are a shorthand way of expressing repeated multiplication of the same number.
For example, instead of writing \( 3 \times 3 \times 3 \times 3 \), we can write \( 3^4 \), which means “3 raised to the power 4”.
Positive Indices
A positive index tells us how many times to multiply a number by itself.
Example: \( 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \)
Zero Index
Any non-zero number raised to the power of zero is equal to 1.
Example: \( 7^0 = 1 \), \( 1000^0 = 1 \)
Negative Indices
A negative index means you take the reciprocal (or “flip”) of the number raised to the positive index.
Example: \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \)
Key Summary
- Positive index: \( a^n = a \times a \times \ldots \times a \) (n times)
- Zero index: \( a^0 = 1 \) for any \( a \neq 0 \)
- Negative index: \( a^{-n} = \frac{1}{a^n} \)
Example :
Evaluate: \( 5^0 \)
▶️ Answer/Explanation
Any number to the power 0 is 1.
Answer: 1
Example :
Evaluate: \( 3^{-2} \)
▶️ Answer/Explanation
\( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)
Answer: \( \frac{1}{9} \)
Example :
Evaluate: \( 2^4 + 2^{-1} \)
▶️ Answer/Explanation
\( 2^4 = 16 \), \( 2^{-1} = \frac{1}{2} \)
\( 16 + \frac{1}{2} = 16.5 \)
Answer: 16.5
Rules of Indices
Understanding the Rules of Indices
The rules (laws) of indices help us simplify expressions involving powers with the same base. These are especially useful when multiplying, dividing, or raising powers to other powers.
1. Product Rule: $a^m \times a^n = a^{m+n}$
When multiplying terms with the same base, add the powers.
Example: \( 2^3 \times 2^4 = 2^{3+4} = 2^7 \)
2. Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$
When dividing terms with the same base, subtract the powers.
Example: \( \frac{5^6}{5^2} = 5^{6-2} = 5^4 \)
3. Power of a Power Rule: $(a^m)^n = a^{m \times n}$
When raising a power to another power, multiply the indices.
Example: \( (3^2)^4 = 3^{2 \times 4} = 3^8 \)
4. Power of a Product: $(ab)^n = a^n b^n$
When raising a product to a power, raise each factor to that power separately.
Example: \( (2x)^3 = 2^3 \times x^3 = 8x^3 \)
5. Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
A quotient raised to a power means you raise both the numerator and denominator to that power.
Example: \( \left(\frac{4}{5}\right)^2 = \frac{4^2}{5^2} = \frac{16}{25} \)
Important:
- The base must be the same to use the product and quotient rules.
- All rules apply to both numerical and algebraic expressions.
- Combine these rules carefully in mixed problems.
Example :
Simplify: \( 3^2 \times 3^{-5} \div 3^{-1} \)
▶️ Answer/Explanation
Step 1: Use product and quotient rules
\( 3^2 \times 3^{-5} = 3^{2 + (-5)} = 3^{-3} \)
\( 3^{-3} \div 3^{-1} = 3^{-3 – (-1)} = 3^{-2} \)
Answer: \( 3^{-2} \)
Example :
Simplify: \( \left(\frac{2x^3}{y^2}\right)^2 \)
▶️ Answer/Explanation
Apply power of a quotient and power of a product rules:
\( \left(\frac{2x^3}{y^2}\right)^2 = \frac{(2x^3)^2}{(y^2)^2} \)
\( = \frac{2^2 \cdot (x^3)^2}{y^{2 \cdot 2}} = \frac{4x^6}{y^4} \)
Answer: \( \frac{4x^6}{y^4} \)
Example :
Simplify: \( (5a^{-2}b)^3 \)
▶️ Answer/Explanation
Apply power of a product rule:
\( (5a^{-2}b)^3 = 5^3 \cdot (a^{-2})^3 \cdot b^3 = 125a^{-6}b^3 \)
Answer: \( 125a^{-6}b^3 \)