Lesson Note
INTRODUCTION
Indices, also known as exponents, are a fundamental concept in mathematics that allow us to express repeated multiplication in a compact form. They are used everywhere, from calculating interest on savings to measuring the growth of populations, and even in everyday technology such as computers and smartphones. Understanding indices helps students see patterns and relationships in numbers, making it easier to solve problems that involve large or small quantities.
In Nigerian classrooms, indices appear in many contexts: simplifying algebraic expressions, solving equations, and applying scientific notation to handle very large or very small measurements in physics and chemistry. Mastering indices also prepares students for higher-level studies in mathematics, engineering, and the sciences, where exponential growth and decay are common.
This lesson will explore the rules that govern indices, illustrate how to apply them through worked examples, and provide practice problems to reinforce understanding.
MAIN CONTENT
Definition of Indices
Indices represent the number of times a base number is multiplied by itself. If a number $a$ is raised to the power $n$, we write $a^{n}$, where $a$ is the base and $n$ is the exponent. The exponent tells us how many copies of $a$ are multiplied together.
The concept of an index is essential because it turns long multiplication into a simple notation. For example, $3^{4}$ means $3 \times 3 \times 3 \times 3$, which equals $81$.
Worked Example
Step 1: Identify the base and exponent: $5^{3}$.
Step 2: Multiply the base by itself three times: $5 \times 5 \times 5$.
Step 3: Calculate: $5 \times 5 = 25$; $25 \times 5 = 125$.
Answer: $5^{3} = 125$.
Practice Questions
1. What is $2^{5}$? Answer: $32$.
2. Evaluate $7^{2}$. Answer: $49$.
3. Compute $10^{3}$. Answer: $1000$.
Basic Laws of Indices
The laws of indices allow us to simplify expressions with exponents. The main laws are:
1. Product Rule: $a^{m} \times a^{n} = a^{m+n}$.
2. Quotient Rule: $\dfrac{a^{m}}{a^{n}} = a^{m-n}$.
3. Power Rule: $(a^{m})^{n} = a^{mn}$.
4. Zero Exponent Rule: $a^{0} = 1$ (for $a \neq 0$).
5. Negative Exponent Rule: $a^{-n} = \dfrac{1}{a^{n}}$.
These rules help in simplifying complex expressions and solving equations efficiently.
Worked Example
Step 1: Simplify $\dfrac{2^{5} \times 2^{3}}{2^{4}}$.
Step 2: Apply the product rule: $2^{5} \times 2^{3} = 2^{5+3} = 2^{8}$.
Step 3: Apply the quotient rule: $\dfrac{2^{8}}{2^{4}} = 2^{8-4} = 2^{4}$.
Step 4: Evaluate: $2^{4} = 16$.
Answer: $16$.
Practice Questions
1. Simplify $3^{2} \times 3^{3}$. Answer: $3^{5} = 243$.
2. Compute $\dfrac{5^{6}}{5^{2}}$. Answer: $5^{4} = 625$.
3. Evaluate $(4^{3})^{2}$. Answer: $4^{6} = 4096$.
Indices with Negative Exponents
A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, $a^{-n} = \dfrac{1}{a^{n}}$. This rule is useful when dealing with fractions or simplifying expressions where the exponent is negative.
Worked Example
Step 1: Rewrite $3^{-2}$ as a fraction: $3^{-2} = \dfrac{1}{3^{2}}$.
Step 2: Compute the denominator: $3^{2} = 9$.
Step 3: Express the fraction: $\dfrac{1}{9}$.
Answer: $3^{-2} = \dfrac{1}{9}$.
Practice Questions
1. What is $5^{-1}$? Answer: $\dfrac{1}{5}$.
2. Simplify $2^{-3}$. Answer: $\dfrac{1}{8}$.
3. Compute $7^{-2}$. Answer: $\dfrac{1}{49}$.
Indices with Fractional Exponents (Roots)
A fractional exponent $\dfrac{m}{n}$ indicates the $n$-th root of the base raised to the $m$-th power. That is, $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$. Common examples include square roots ($\frac{1}{2}$) and cube roots ($\frac{1}{3}$).
Worked Example
Step 1: Evaluate $\sqrt[3]{8^{2}}$.
Step 2: Compute the power inside the root: $8^{2} = 64$.
Step 3: Find the cube root of 64: $\sqrt[3]{64} = 4$.
Answer: $4$.
Practice Questions
1. Compute $\sqrt{25}$. Answer: $5$.
2. Evaluate $\sqrt[4]{81}$. Answer: $3$.
3. Find $\sqrt[3]{27^{2}}$. Answer: $9$.
Indices of Zero and One
The exponent zero rule states that any non-zero base raised to the zero power equals one: $a^{0} = 1$. The exponent one rule states that any base raised to the first power equals the base itself: $a^{1} = a$. These rules help simplify expressions and avoid unnecessary calculations.
Worked Example
Step 1: Simplify $9^{0}$.
Step 2: Apply the zero exponent rule: $9^{0} = 1$.
Answer: $1$.
Step 1: Simplify $7^{1}$.
Step 2: Apply the one exponent rule: $7^{1} = 7$.
Answer: $7$.
Practice Questions
1. What is $4^{0}$? Answer: $1$.
2. Evaluate $12^{1}$. Answer: $12$.
3. Compute $0^{0}$? Answer: Indeterminate (commonly defined as $1$ in many contexts but requires caution).
Indices in Scientific Notation
Scientific notation expresses very large or very small numbers as a product of a number between 1 and 10 and a power of ten. For example, $3.2 \times 10^{5}$ represents $320\,000$. Indices are crucial in scientific notation because they allow us to handle large exponents efficiently.
Worked Example
Step 1: Express $0.00056$ in scientific notation.
Step 2: Move the decimal point five places to the right: $5.6$.
Step 3: Count the moves: $5$ places.
Step 4: Write the number: $5.6 \times 10^{-5}$.
Answer: $5.6 \times 10^{-5}$.
Practice Questions
1. Write $8\,000$ in scientific notation. Answer: $8 \times 10^{3}$.
2. Express $0.0034$ in scientific notation. Answer: $3.4 \times 10^{-3}$.
3. Convert $2.5 \times 10^{6}$ to a standard number. Answer: $2\,500\,000$.
Application of Indices in Problem Solving
Indices are used to solve real-world problems such as calculating compound interest, determining population growth, and measuring the decay of radioactive substances. By applying the laws of indices, students can set up and solve equations that model these situations.
Worked Example
A savings account offers a 5% annual interest rate, compounded annually. If a student deposits $$100 naira, how much will be in the account after 4 years? Step 1: Identify the principal $$P = 100$ naira, the rate $r = 0.05$, and the number of years $n = 4$$. Step 2: Use the compound interest formula: $$A = P(1 + r)^{n}$. Step 3: Substitute values: $A = 100(1 + 0.05)^{4}$. Step 4: Compute the base: $1 + 0.05 = 1.05$. Step 5: Raise to the power of 4: $(1.05)^{4} \approx 1.21550625$. Step 6: Multiply by the principal: $100 \times 1.21550625 \approx 121.55$. Answer: Approximately $121.55$ naira. Practice Questions 1. If $2^{3} \times 2^{2}$$ is simplified, what is the result? Answer: $$2^{5} = 32$. 2. Compute $5^{-2} \times 25$. Answer: $1$. 3. Evaluate $\sqrt[3]{27^{2}} \times 3$. Answer: $27$$. SUMMARY Indices provide a powerful shorthand for repeated multiplication and are foundational to algebra, calculus, and many applied sciences. By mastering the basic laws—product, quotient, power, zero, and negative exponents—students can simplify expressions, solve equations, and model real-world phenomena. Indices also enable the use of scientific notation, allowing us to handle extremely large or small numbers with ease. The worked examples and practice problems in this lesson reinforce these concepts, ensuring students can apply indices confidently in both academic and everyday contexts. EVALUATION 1. Simplify $$4^{2} \times 4^{3}$. 2. Evaluate $\dfrac{9^{5}}{9^{2}}$. 3. Compute $6^{-3}$. 4. Find $\sqrt[4]{16^{3}}$. 5. Express $0.00012$ in scientific notation.