What is Index Notation?

Index notation is a shorthand way of writing repeated multiplication of the same number. For example, instead of writing 2 × 2 × 2, we can write 2³.

The small number (in this case, 3) is called the index or exponent or power. The number being multiplied (in this case, 2) is called the base. 2³ is read as "two to the power of three" or "two cubed."

Let’s go through some very simple questions (Please watch the video for detailed workings to get to the solutions) :

1. Write 3 × 3 × 3 × 3 × 3 in index notation.
2. What is the base in the expression 4⁶?
3. What is the index in the expression 10²?
4. Write 8³ in expanded form (without calculating the final answer).
5. Express 6 × 6 × 6 × 6 × 6 × 6 in index notation.

Solutions:

1. 3⁵
2. 4
3. 2
4. 8 × 8 × 8
5. 6⁶

Index Laws: Multiplication, Division, Power of a Power, Zero Index

Here are some rules you need to know when you’re working with indices.

• Multiplication - To multiply numbers with the same base, keep the base and add the indices: a^m × aⁿ = a^m+n
• Division - To divide numbers with the same base, keep the base and subtract the indices: a^m ÷ aⁿ = a^m-n
• Power of a Power Law - To raise a power to another power, keep the base and multiply the indices: (a^m)ⁿ = a^m×n
• Zero Index - Any non-zero number raised to the power of zero is 1: a⁰ = 1 (where a ≠ 0)

Let’s go through some questions:

1. Simplify 7¹ × 7³.
2. If 2^x × 2⁵ = 2⁸, what is the value of x?
3. Simplify 3¹⁰ ÷ 3⁶.
4. If 2⁷ ÷ 2^x = 2⁴, what is the value of x?
5. Simplify (7³)³.
6. If (2^x)³ = 2¹², what is the value of x?
7. Simplify 9² ÷ 9² using the division law and then using the zero index law.
8. Simplify 3⁵ × 3² ÷ 3⁷.

Solutions:

1. 7⁴
2. 3
3. 3⁴
4. 3
5. 7⁹
6. 4
7. 9^2-2 = 9⁰ = 1
8. 3^5+2-7 = 3⁰ = 1

Index Laws in Application

So far, you've mastered the basics of index notation and learned the important index laws. You can confidently simplify expressions using these rules. However, this exam isn't going to simply hand you a formula like a^m × aⁿ = a^m+n and ask you to plug in numbers.

Instead, you need to be able to recognize when index notation is the right tool to use. The real challenge lies in analysing a worded problem, often disguised in a real-world scenario, and figuring out how to apply index notation to solve it.

This means you'll need to decide which numbers represent the base and which represent the index, how to set up the equation yourself, and ultimately, how to use your knowledge of index laws to find any missing information the problem is asking for.

Essentially, you are expected to be able to determine if it is an index notation question, and then decide how to correctly write the equation to solve the problem. You are not just being tested on your ability to calculate, but your ability to apply mathematical thinking.

Let’s see some examples of more difficult questions in action

Question 1: The Spreading Secret

Sarah tells a secret to three friends. Each of those friends tells the secret to three more friends the next day, and so on. The number of people who hear the secret each day follows a pattern: 3, 9, 27, and so on.

If this pattern continues, on what day, represented by 'd', will 729 people hear the secret? Express the number of people who hear the secret on day 'd' using index notation.

Thought Process:

Identify the pattern: The number of people hearing the secret triples each day (multiplied by 3). This indicates a base of 3 in index notation.

Relate the pattern to days:

• Day 1: 3 people (3¹)
• Day 2: 9 people (3²)
• Day 3: 27 people (3³)

Notice the day number matches the exponent.

Solve for 'd': We need to find the exponent (day number) that makes 3 raised to that power equal to 729. In other words, find 'd' where 3^d = 729.

Calculate: 3⁴ = 81, 3⁵ = 243, 3⁶ = 729.

Answer:

'd' (the day number) is 6.

The number of people who hear the secret on day 'd' (day 6) is 729, which in index notation is 3^d, or 3⁶

Question 2: The Viral Video

A funny cat video gets uploaded to the internet. On the first day, it gets 2 views. Each day after that, the number of views doubles.

After 'n' days, the video has 128 views in total. How many days has the video been online? Also, express the total views on day 'n' using index notation.

Thought Process:

Identify the pattern: The number of views doubles each day (multiplied by 2). This suggests a base of 2 in index notation.

Relate the pattern to days:

• Day 1: 2 views (2¹)
• Day 2: 4 views (2²)
• Day 3: 8 views (2³)

The day number matches the exponent.

Solve for 'n': We need to find the exponent 'n' such that 2ⁿ = 128.

Calculate: 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128.

Answer:

'n' (the number of days) is 7.

The total views on day 'n' (day 7) are 128, expressed in index notation as 2ⁿ, or 2⁷

Practice time!

Now, it's your turn to practice.

The questions in this checkpoint are provided to give you an introduction to possible questions you may see in your exam. Don't worry too much as you'll continue to build your skills throughout the course.

Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.

Every question has a suggested solutions videos after you complete the question. This video explains to you the steps to take to answer the question and provides tips and tricks.

Once you're done with the practice questions, move on to the next checkpoint.

Now, let’s get started on your practice questions.

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