**This Mathematics Tutorial will focus on INDICES. We will explain it with examples and give you exercise. At the end of the tutorial, you’ll be able to download it for FREE. Please share this page with your friends who may need it.**

In mathematics, **indices**, also known as exponents or powers, are a way of expressing the repeated multiplication of a number by itself.

We will break this down for you.

Let’s say you have a number like 2, and you want to multiply it by itself a certain number of times. Instead of writing out the multiplication several times, you can use indices to represent this more concisely.

Here’s how it works:

**Base Number:**The number you’re multiplying by itself is called the base number. In our example, the base number is 2.**Exponent:**The small number written above and to the right of the base number is called the exponent. It tells you how many times to multiply the base number by itself.

Let me give you an example:

- 2
^{3}

Here, 2 is the **base number**, and **3 is the exponent**. It means you multiply 2 by itself **three times**:

2^{3} = 2Ã—2Ã—2 = 8

So, 2^{3} is **equal to 8**.

Let’s try another one:

- 5
^{2}

Here, **5** is the **base number**, and **2** is the **exponent**. It means you multiply **5** by itself **two times**:

5^{2}= 5Ã—5 =25

So, 5^{2} is equal to 25.

Now let’s look at an example with a **negative **base:

- (âˆ’3)
^{2}

Here, **-3** is the **base number**, and **2** is the **exponent**. It means you **multiply -3** **by itself two times**:

(âˆ’3)^{2} = (âˆ’3) Ã— (âˆ’3) = 9

Notice that when we multiply two negative numbers, we get a positive result.

Also note that: A negative base raised to an odd index is negative. While a negative base raised to an even index is positive

Now, let’s look at a case with a **negative exponent**:

- 2
^{âˆ’3}

Here, **2 is the base number**, and **-3 is the exponent**. It means you take **the reciprocal of 2 raised to the positive 3**:

2^{âˆ’3}= Â½Â³ = â…›

In this case, a negative exponent indicates that we take the reciprocal^{1} of the base number raised to the positive value of the exponent.

**Laws of Indices**

The laws of indices, also known as the laws of exponents, are a set of rules that help simplify expressions involving powers. Here are the main laws of indices with examples:

**Product Law:**a^{m}Ã— a^{n}= a^{m+n}

This law states that when you multiply two terms with the same base, you add the exponents.**Example:**2^{3}Ã—2^{4}= 2^{3+4}= 2^{7}=128**Quotient Law:**a^{m}/ a^{n}=a^{mâˆ’n}

This law states that when you divide two terms with the same base, you subtract the exponents.**Example:**3^{2}/ 3^{5}= 3^{5âˆ’2}= 3^{3}=27**Power Law:**(a^{m})^{n}=a^{mâˆ’n}

This law states that when you raise a power to another power, you multiply the exponents.**Example:**(4^{2})^{3}=4^{2Ã—3}=4^{6}=4096**Negative Exponent Law:**a^{âˆ’m}= 1 / a^{m}

This law states that a negative exponent is equivalent to the reciprocal of the term with a positive exponent.**Example:**2^{âˆ’3}=1 / 2^{3}= 1 / 8**Zero Exponent Law:**a^{0}= 1

Any nonzero number raised to the power of zero is equal to 1.**Example:**5^{0}= 1

These laws are fundamental in simplifying expressions involving powers and provide a systematic way to manipulate and solve problems involving exponents.

THINK & ACT :If we can give you this for FREE, imagine what we can give if you pay and join theALLSCHOOL JAMB Online Lesson. In the lesson, ourhardworking tutorsensure they onlyteach you things that will come out in JAMB, so you’llscore extremely high in JAMB. Click Here to join theALLSCHOOL JAMB Online LessonNOW.

**Exercises**:

(1) List the **first six powers **of (a) 2 (b) 3 (c) 4

(2) Copy and complete the values of these common powers

(a) 5^{1}

(b) 5^{4}

(c) 6^{3}

(d) 5^{2}

(e) 6^{1}

(f) 7^{4}

(3) Simplify

(a) (-1)^{5}

(b) (-1)^{6}

(c) (-2)^{5}

(4) Simplify:

(a) 4^{x} = 8

(b) 9^{x-2} = â…“

## SOLUTIONS TO EXERCISES

Make sure you try solving the exercises yourself. It’ll really help you understand the topic.

(1) List the **first six powers **of (a) 2 (b) 3 (c) 4

**(a) First six powers of 2:**

- 2
^{1}=2 - 2
^{2}=4 - 2
^{3}=8 - 2
^{4}=16 - 2
^{5}=32 - 2
^{6}=64

**(b) ****First six p**owers of 3:

- 3
^{1}=3 - 3
^{2}=9 - 3
^{3}=27 - 3
^{4}=81 - 3
^{5}=243 - 3
^{6}=729

**(c) ****First six p**owers of 4:

- 4
^{1}=4 - 4
^{2}=16 - 4
^{3}=64 - 4
^{4}=256 - 4
^{5}=1024 - 4
^{6}=4096

**(2) Copy and complete the values of these common powers **

(a) 5^{1}

(b) 5^{4}

(c) 6^{3}

(d) 5^{2}

(e) 6^{1}

(f) 7^{4}

**(a) 5 ^{1}:** 5 x 1 = 5

**(b) 5 ^{4}:** 5Ã—5Ã—5Ã—5=625

**(c) 6 ^{3}: **6Ã—6Ã—6=216

**(d) 5 ^{2}:** 5Ã—5=25

**(e) 6 ^{1}:** 6Ã—1=6

**(f) 7 ^{4}:** 7Ã—7Ã—7Ã—7=2401

These values represent the results of raising the given bases to the specified powers using indices.

**(3) Simplify ** (a) (-1)^{5} (b) (-1)^{6} (c) (-2)^{5}

**(a) (âˆ’1) ^{5}** means multiplying -1 by itself 5 times. ie (âˆ’1)

^{5}= âˆ’1 Ã— âˆ’1 Ã— âˆ’1 Ã— âˆ’1 Ã—âˆ’1 = âˆ’1

**(b) (âˆ’1) ^{6}** means multiplying -1 by itself 6 times. (âˆ’1)

^{6}= âˆ’1Ã— âˆ’1Ã— âˆ’1Ã— âˆ’1Ã— âˆ’1Ã— âˆ’1 = 1

**(c) (âˆ’2)** means multiplying -2 by itself 5 times. (âˆ’2)^{5} = âˆ’2Ã— âˆ’2Ã— âˆ’2Ã— âˆ’2Ã— âˆ’2 = âˆ’32

So, after simplifying:

(a) (âˆ’1)^{5} = âˆ’1,

(b) (âˆ’1)^{6} = 1,

(c) (âˆ’2)^{5} = âˆ’32.

(4) Simplify:

(a) 4^{x} = 8

(b) 9^{x-2} = â…“

**(a) 4 ^{x} = 8**

2

^{2x}= 2

^{3}

Equate the powers, therefore

2x = 3,

x = 3/2 or 1Â½

**(b) 9 ^{x-2} = â…“**

(3

^{2})

^{x-2}= 3

^{-1}

Equate the powers,

2(x-2) = -1,

2x-4 = -1,

2x = -1+4

2x = 3,

x = 3/2

## Download Maths Tutorial for Free

**Footnotes**:

- The reciprocal of a number is simply 1/number.

So, if you have a number like 2, the reciprocal of 2 is Â½â€‹.

Similarly, the reciprocal of âˆ’3 is 1/âˆ’3â€‹.

In the case of 2^{âˆ’3}, it means 1/2^{3}â€‹, which is â…›â€‹.

In summary,**reciprocals**express the multiplicative inverse of a number, meaning a number that, when multiplied by the original number, gives a product of 1. â†©ï¸Ž

JOIN NOW: In theALLSCHOOL JAMB Online Lesson, our teachers focus on the exact things that will most likely come out in JAMB. Joining the Lesson is one of the surest ways to score high in JAMB. The lesson is a live-stream class. Click here to see how to join and other details.

**RECOMMENDED ARTICLES FOR JAMB ASPIRANTS**

- Join Allschool JAMB Lesson and Score Very High in JAMB [Tested & Trusted]
- Practice Real JAMB Past Questions with ALLSCHOOL JAMB CBT App
- Install LitTexts App:
*2024 Literature Prose, Drama, & Poetry at your fingertips*. - How to Score Extremely High In JAMB [
**7 Trusted Tips**] - JAMB Free Tutorials [ðŸ’¯ Free]
- All JAMB News and Updates
- Download JAMB 2024 Novel – The Life Changer [
**Free PDF**] - Likely Questions And Answers From The Life Changer
- When is the JAMB Exam Starting?
- JAMB Runs is FAKE
- Idioms You Might See in Your English JAMB Questions [
**Hurry, Check Them Out!**] - JAMB Syllabus for All Subjects [
**Official & Updated**] - Updated Accredited JAMB CBT Centers in Nigeria.
- How to Buy 2024 JAMB Form [Everything you need to know]
- How Much Is JAMB Form 2024
- Solutions to Profile Code Issues [Get Yours Fixed Instantly]
- How to Fill JAMB Form the Right Way [Avoid Silly Mistakes]
- JAMB Brochure: JAMB Subject Combination for all Courses

ALLSCHOOL TEAM

**Thank you so much for reading. We will appreciate it if you share this with your loved ones.**