Table of Contents
What are Indices?
Indices (also called powers or exponents) are a way of writing repeated multiplication in a shorter form. For example, instead of writing $2 \times 2 \times 2 \times 2$, we can write $2^4$, where 4 is the index (or power) and 2 is the base.
Understanding the Parts: #
$\underbrace{2}_{\text{base}} \text{ }^{\overbrace{4}^{\text{index/power}}}$
- Base: The number being multiplied repeatedly
- Index/Power: How many times we multiply the base by itself
- Reading: $2^4$ is read as “2 to the power of 4” or “2 to the 4th”
Why Do We Use Indices? #
- Shorter writing: $2^{10}$ instead of $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
- Easier calculations: We can use rules instead of doing long multiplication
- Scientific notation: For very large or small numbers like $3.2 \times 10^8$
6. Step-by-Step Problem Solving Strategy #
When faced with complex index problems, follow this step-by-step approach:
Problem-Solving Steps: #
- Identify what you have: Look for bases, powers, fractions, negatives
- Break down complex expressions: Deal with brackets first
- Apply one rule at a time: Don’t try to do everything at once
- Simplify step by step: Write down each step clearly
- Check your answer: Does it make sense?
Complex Example: Simplify $\frac{(2^3)^2 \times 2^{-1}}{2^4}$
Step 1: Deal with the bracket first: $(2^3)^2 = 2^{3 \times 2} = 2^6$
Step 2: Now we have: $\frac{2^6 \times 2^{-1}}{2^4}$
Step 3: Multiply the numerator: $2^6 \times 2^{-1} = 2^{6+(-1)} = 2^5$
Step 4: Now we have: $\frac{2^5}{2^4}$
Step 5: Apply division rule: $\frac{2^5}{2^4} = 2^{5-4} = 2^1 = 2$
Answer: $2$
Common Student Mistakes and How to Avoid Them: #
- Mistake: $2^3 + 2^4 = 2^7$ ❌
Correct: $2^3 + 2^4 = 8 + 16 = 24$ (can’t combine different operations) - Mistake: $3^2 \times 5^2 = 15^4$ ❌
Correct: $3^2 \times 5^2 = 9 \times 25 = 225$ or $(3 \times 5)^2 = 15^2$ - Mistake: $(x + y)^2 = x^2 + y^2$ ❌
Correct: $(x + y)^2 = x^2 + 2xy + y^2$ (expand the bracket) - Mistake: $\sqrt{x^2 + y^2} = x + y$ ❌
Correct: $\sqrt{x^2 + y^2}$ cannot be simplified further
1.5. Power of a Power Rule #
Rule:
$(a^m)^n = a^{mn}$When raising a power to another power, multiply the indices.
Simple Memory Tip: #
Power of a power: multiply the powers together.
Quick Examples: #
- $(x^2)^5 = x^{2 \times 5} = x^{10}$
- $(3^4)^2 = 3^{4 \times 2} = 3^8$
- $(a^{-2})^3 = a^{-2 \times 3} = a^{-6}$
Worked Example: Simplify $(5^2)^3 \times 5^4$
Step 1: Deal with the power of a power first: $(5^2)^3 = 5^{2 \times 3} = 5^6$
Step 2: Now we have: $5^6 \times 5^4$
Step 3: Use multiplication rule: $5^6 \times 5^4 = 5^{6+4} = 5^{10}$
Answer: $5^{10}$
1. Basic Index Rules #
There are several important rules for working with indices. These rules work because of what indices actually mean (repeated multiplication). Learning these rules will help you simplify expressions and solve problems more easily.
Important Note: #
The rules for indices ONLY work when the bases are the same. For example, $3^2 \times 3^4$ can use the rules, but $3^2 \times 5^4$ cannot because the bases (3 and 5) are different.
Rule 1: Multiplying Powers with the Same Base #
Rule:
$a^m \times a^n = a^{m+n}$When multiplying powers with the same base, add the indices.
Simple Memory Tips: #
- Same base multiplication: Keep the base, add the powers
- Same base division: Keep the base, subtract the powers
- Different bases: You cannot combine them with index rules
Quick Examples: #
- $3^2 \times 3^4 = 3^{2+4} = 3^6$
- $5^8 \div 5^3 = 5^{8-3} = 5^5$
- $3^2 \times 5^4$ cannot be simplified (different bases)
Worked Example from Question 1(a): Simplify $h^2 \times h^3$
Step 1: Identify the base and powers: base = $h$, powers = 2 and 3
Step 2: Apply the rule: $h^2 \times h^3 = h^{2+3}$
Step 3: Add the powers: $h^{2+3} = h^5$
Answer: $h^5$
Rule 2: Negative Powers #
Rule:
$a^{-n} = \frac{1}{a^n}$A negative power means “1 divided by that positive power”.
Simple Rule for Negative Powers: #
- For numbers: $a^{-n} = \frac{1}{a^n}$ (put it under 1, make power positive)
- For fractions: Flip the fraction upside down and make the power positive
Quick Examples: #
- $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
- $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$
- $\left(\frac{7}{8}\right)^{-1} = \frac{8}{7}$ (just flip it)
Worked Example from Question 1(b): Simplify $\left(\frac{7}{8}\right)^{-1}$
Step 1: Apply the negative power rule: $\left(\frac{7}{8}\right)^{-1} = \frac{8}{7}$ (flip the fraction)
Answer: $\frac{8}{7}$
Rule 3: Dividing Powers with the Same Base #
Rule:
$a^m \div a^n = a^{m-n}$When dividing powers with the same base, subtract the indices.
Simple Memory Tip: #
When dividing same bases: subtract the bottom power from the top power.
Quick Examples: #
- $7^8 \div 7^3 = 7^{8-3} = 7^5$
- $x^{10} \div x^7 = x^{10-7} = x^3$
- $5^4 \div 5^4 = 5^{4-4} = 5^0 = 1$
- $3^2 \div 3^5 = 3^{2-5} = 3^{-3} = \frac{1}{3^3}$
Worked Example from Question 1(c): $a^8 \div a^p = a^2$, find the value of $p$
Step 1: Apply the division rule: $a^8 \div a^p = a^{8-p}$
Step 2: We know this equals $a^2$, so: $a^{8-p} = a^2$
Step 3: For the powers to be equal: $8 – p = 2$
Step 4: Solve for $p$: $p = 8 – 2 = 6$
Answer: $p = 6$
Worked Example from Question 1(d): Find the value of $p$ when $6^p \times 6^4 = 6^{28}$
Step 1: Apply the multiplication rule: $6^p \times 6^4 = 6^{p+4}$
Step 2: We know this equals $6^{28}$, so: $6^{p+4} = 6^{28}$
Step 3: For the powers to be equal: $p + 4 = 28$
Step 4: Solve for $p$: $p = 28 – 4 = 24$
Answer: $p = 24$
2. Standard Form and Powers of 10 #
Simple Rules for Standard Form: #
- Large numbers: Move decimal left → positive power
- Small numbers: Move decimal right → negative power
- First number: Must be between 1 and 10
Quick Examples: #
- $45,000 = 4.5 \times 10^4$ (moved 4 places left)
- $0.0067 = 6.7 \times 10^{-3}$ (moved 3 places right)
Worked Example from Question 2(a): Work out $2 \times 10^{100} – 2 \times 10^{98}$
Step 1: Factor out the common term: $2 \times 10^{98}(10^2 – 1)$
Step 2: Calculate: $10^2 – 1 = 100 – 1 = 99$
Step 3: So we have: $2 \times 10^{98} \times 99 = 198 \times 10^{98}$
Step 4: Convert to standard form: $1.98 \times 10^{100}$
Answer: $1.98 \times 10^{100}$
Worked Example from Question 2(b): Work out $\frac{6.39 \times 10^4}{2.45 \times 10^6}$
Step 1: Separate the numbers and powers: $\frac{6.39}{2.45} \times \frac{10^4}{10^6}$
Step 2: Calculate the numbers: $\frac{6.39}{2.45} \approx 2.61$
Step 3: Calculate the powers: $\frac{10^4}{10^6} = 10^{4-6} = 10^{-2}$
Step 4: Combine: $2.61 \times 10^{-2}$
Answer: $2.61 \times 10^{-2}$
3. Fractional and Complex Indices #
Simple Rules for Fractional Powers: #
- Bottom number = which root: $x^{\frac{1}{2}} = \sqrt{x}$, $x^{\frac{1}{3}} = \sqrt[3]{x}$
- Top number = regular power
- Easy method: Find the root first, then raise to the power
Quick Examples: #
- $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$
- $16^{\frac{1}{4}} = \sqrt[4]{16} = 2$
- $9^{\frac{3}{2}} = (\sqrt{9})^3 = 3^3 = 27$
- $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$
Fractional Indices #
Rules:
$a^{\frac{1}{n}} = \sqrt[n]{a}$ $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$A fractional index represents a root.
Understanding $a^{\frac{m}{n}}$: #
Easy method: Root first, then power → $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$
Examples: #
- $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$
- $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$
- $16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8$
Worked Example from Question 3: Simplify $\left(216^{\frac{2}{6}}\right)^{\frac{1}{2}}$
Step 1: Simplify the inner power: $\frac{2}{6} = \frac{1}{3}$
Step 2: So we have: $\left(216^{\frac{1}{3}}\right)^{\frac{1}{2}}$
Step 3: $216^{\frac{1}{3}} = \sqrt[3]{216} = 6$ (since $6^3 = 216$)
Step 4: Now we have: $6^{\frac{1}{2}} = \sqrt{6}$
Answer: $\sqrt{6}$
Worked Example from Question 3: Simplify $\left(3125x^{-3125}\right)^{\frac{1}{5}}$
Step 1: Apply the power to each part: $3125^{\frac{1}{5}} \times \left(x^{-3125}\right)^{\frac{1}{5}}$
Step 2: Find the fifth root of 3125: $3125^{\frac{1}{5}} = 5$ (since $5^5 = 3125$)
Step 3: Apply the power rule: $\left(x^{-3125}\right)^{\frac{1}{5}} = x^{-3125 \times \frac{1}{5}} = x^{-625}$
Step 4: Combine: $5x^{-625} = \frac{5}{x^{625}}$
Answer: $5x^{-625}$ or $\frac{5}{x^{625}}$
Worked Example from Question 3: Simplify $18x^{18} + 9x^9$
Step 1: Factor out the common factors: $9x^9(2x^9 + 1)$
Step 2: Check: $9x^9 \times 2x^9 = 18x^{18}$ ✓
Step 3: Check: $9x^9 \times 1 = 9x^9$ ✓
Answer: $9x^9(2x^9 + 1)$
4. Zero and Negative Powers #
Simple Rule for Zero Powers: #
Any number to the power of 0 equals 1 (except $0^0$ which is undefined)
Quick Examples: #
- $5^0 = 1$
- $(-3)^0 = 1$
- $x^0 = 1$ (as long as $x \neq 0$)
- $7x^0 = 7 \times 1 = 7$
Zero Power Rule #
Rule:
$$a^0 = 1 \text{ (where } a \neq 0\text{)}$$Any number (except zero) raised to the power of 0 equals 1.
Worked Example from Question 4:
(a) $y^3 \div y^3 = y^{3-3} = y^0 = 1$
(b) $7x^0 = 7 \times 1 = 7$ (since $x^0 = 1$)
More Complex Fractional Indices #
Worked Example from Question 5:
(a) Simplify $\left(81x^{16}\right)^{\frac{3}{4}}$
Step 1: Apply the power to each part: $81^{\frac{3}{4}} \times \left(x^{16}\right)^{\frac{3}{4}}$
Step 2: $81^{\frac{3}{4}} = \left(81^{\frac{1}{4}}\right)^3 = 3^3 = 27$
Step 3: $\left(x^{16}\right)^{\frac{3}{4}} = x^{16 \times \frac{3}{4}} = x^{12}$
Answer: $27x^{12}$
(b) Simplify $\left(\frac{1}{y^2}\right)^{-\frac{1}{2}}$
Step 1: Use the rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$
Step 2: $\left(\frac{1}{y^2}\right)^{-\frac{1}{2}} = \left(\frac{y^2}{1}\right)^{\frac{1}{2}} = \left(y^2\right)^{\frac{1}{2}}$
Step 3: $\left(y^2\right)^{\frac{1}{2}} = y^{2 \times \frac{1}{2}} = y^1 = y$
Answer: $y$
5. Key Points to Remember #
- Same base multiplication: $a^m \times a^n = a^{m+n}$
- Same base division: $a^m \div a^n = a^{m-n}$
- Power of a power: $(a^m)^n = a^{mn}$
- Zero power: $a^0 = 1$ (where $a \neq 0$)
- Negative power: $a^{-n} = \frac{1}{a^n}$
- Fractional power: $a^{\frac{1}{n}} = \sqrt[n]{a}$
- Standard form: $a \times 10^n$ where $1 \leq a < 10$
- Power of a product: $(ab)^n = a^n b^n$
- Power of a quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
Common Mistakes to Avoid:
- Don’t add indices when the bases are different
- Remember that $0^0$ is undefined
- Be careful with negative signs in indices
- When converting to standard form, make sure the first number is between 1 and 10
- $(a + b)^n \neq a^n + b^n$ (this is a common error!)
