Index Laws
Understanding and applying the rules of exponents
๐ Introduction
Indices (or powers) provide a concise way to express repeated multiplication. Instead of writing $2 \times 2 \times 2 \times 2 \times 2$, we can simply write $2^5$. The number $2$ is the base, and $5$ is the index (or exponent).
Index laws are the rules that allow us to manipulate expressions with powers. These laws make calculations involving powers much simpler and are essential for working with algebraic expressions, scientific notation, and many areas of mathematics.
๐ฏ Learning Objectives
By the end of this lesson, you will be able to:
- Understand what indices represent and how to interpret them
- Apply the laws of indices to simplify expressions
- Evaluate expressions with indices correctly
- Solve problems involving multiple index laws
๐ Key Concepts
Understanding Indices
An index (plural: indices) is a small number written above and to the right of another number or variable, indicating how many times that number or variable is multiplied by itself.
๐ Definition
Index notation: For a positive integer $n$, $$a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ times}}$$ where $a$ is the base and $n$ is the index/exponent.
The Laws of Indices
There are several key laws that govern how indices behave in mathematical operations. Let's explore each one:
๐ Law 1: Multiplication Law
When multiplying powers with the same base, add the indices: $$a^m \times a^n = a^{m+n}$$
Example: $2^3 \times 2^4 = 2^{3+4} = 2^7$
๐ Law 2: Division Law
When dividing powers with the same base, subtract the indices: $$\frac{a^m}{a^n} = a^{m-n}$$
Example: $\frac{3^5}{3^2} = 3^{5-2} = 3^3$
๐ Law 3: Power of a Power Law
When raising a power to another power, multiply the indices: $$(a^m)^n = a^{m \times n}$$
Example: $(2^3)^4 = 2^{3 \times 4} = 2^{12}$
๐ Law 4: Zero Power Law
Any non-zero number raised to the power of zero equals 1: $$a^0 = 1 \quad \text{(where } a \neq 0 \text{)}$$
Example: $7^0 = 1$, $x^0 = 1$ (where $x \neq 0$)
๐ก Pro Tip
Remember that these laws only work when the bases are the same. For example, $2^3 \times 3^2$ cannot be simplified using the index laws because the bases (2 and 3) are different.
๐ Worked Examples
๐ Example 1: Applying the Multiplication Law
Question: Simplify $x^4 \times x^7$.
Solution:
Identify which index law to use. Since we're multiplying two powers with the same base, we'll use the multiplication law.
Apply the multiplication law: $a^m \times a^n = a^{m+n}$
Calculate the sum of the indices.
The simplified expression is:
๐ Example 2: Applying Multiple Index Laws
Question: Simplify $(y^3)^4 \times \frac{y^5}{y^2}$.
Solution:
Break down the problem and tackle each part separately. First, simplify $(y^3)^4$ using the power of a power law.
Next, simplify $\frac{y^5}{y^2}$ using the division law.
Now multiply the two simplified expressions using the multiplication law.
The simplified expression is:
โ ๏ธ Common Mistakes
๐ซ Mistake 1: Adding Indices When Multiplying Different Bases
What students often do wrong: Writing $2^3 \times 3^4 = 5^7$.
Why it's wrong: Index laws only apply when the bases are the same. You cannot add indices for different bases.
How to avoid it: Always check if the bases are the same before applying any index law.
๐ซ Mistake 2: Misapplying the Power of a Power Law
What students often do wrong: Writing $(2 \times 3)^4 = 2^4 \times 3^4$.
Why it's wrong: $(2 \times 3)^4 = 6^4$, which is different from $2^4 \times 3^4$.
How to avoid it: Remember that $(a \times b)^n = a^n \times b^n$, but this is different from $(a^m)^n = a^{m \times n}$.
๐ซ Mistake 3: Confusion with Zero Power
What students often do wrong: Thinking that $a^0 = 0$ or that $0^0$ is defined.
Why it's wrong: $a^0 = 1$ for any non-zero value of $a$, and $0^0$ is indeterminate in standard mathematics.
How to avoid it: Remember that $a^0 = 1$ is a fundamental rule of indices, but it applies only when $a \neq 0$.
๐ช Practice Problems
Try these problems to test your understanding:
๐ฏ Practice Question 1
Simplify the following expressions:
(a) $3^4 \times 3^5$
(b) $\frac{5^7}{5^3}$
(c) $(2^3)^2$
๐ Show Solution
(a) $3^4 \times 3^5 = 3^{4+5} = 3^9 = 19,683$
(b) $\frac{5^7}{5^3} = 5^{7-3} = 5^4 = 625$
(c) $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$
๐ฏ Practice Question 2
Simplify $\frac{x^5 \times x^3}{x^2}$.
๐ Show Solution
Step 1: Simplify the numerator first.
$x^5 \times x^3 = x^{5+3} = x^8$
Step 2: Divide by $x^2$.
$\frac{x^8}{x^2} = x^{8-2} = x^6$
Answer: $\boxed{x^6}$
๐ฏ Practice Question 3
Simplify $(a^2b^3)^4$.
๐ Show Solution
Step 1: Apply the power of a power law to each part.
$(a^2)^4 = a^{2 \times 4} = a^8$
$(b^3)^4 = b^{3 \times 4} = b^{12}$
Step 2: Combine the results.
$(a^2b^3)^4 = a^8b^{12}$
Answer: $\boxed{a^8b^{12}}$
๐ Challenge Exam Question
๐ Exam-style Question:
Simplify the following expression, giving your answer with positive indices only: $$\frac{(x^3y^2)^4 \times (xy^3)^2}{(x^2y)^3}$$
๐ Show Solution
Apply the power of a power law to expand the terms in the numerator.
Expand the term in the denominator.
Combine the terms in the numerator using the multiplication law.
Divide the numerator by the denominator using the division law.
๐ Summary
๐ฏ Key Takeaways
- Multiplication Law: $a^m \times a^n = a^{m+n}$ - Add powers when multiplying terms with the same base.
- Division Law: $\frac{a^m}{a^n} = a^{m-n}$ - Subtract powers when dividing terms with the same base.
- Power of a Power Law: $(a^m)^n = a^{m \times n}$ - Multiply powers when raising a power to another power.
- Zero Power Law: $a^0 = 1$ (where $a \neq 0$) - Any non-zero number raised to the power of zero equals 1.
- These laws only work when the bases are the same.
๐ What's Next?
Now that you have mastered the laws of indices, you're ready to learn about Expanding Brackets, where you'll apply these skills to more complex algebraic expressions. Continue to the next lesson to further develop your algebraic skills.