Exponents and Powers Class 8 Mathematics — Revision Notes

TL;DR: Exponents and powers provide a compact way to express repeated multiplication. They are fundamental to algebra, science, and technology.

Exponents and powers provide a compact way to express repeated multiplication. They are fundamental to algebra, science, and technology.

Basic Concepts

• Exponent or power is the number of times a base is multiplied by itself
• In 2 to the power 5, 2 is the base and 5 is the exponent
• 2 to the power 5 equals 2 times 2 times 2 times 2 times 2 equals 32
• Any non-zero number to the power 0 equals 1
• Any number to the power 1 equals that number

Laws of Exponents

• Product rule: a to power m times a to power n equals a to power m plus n
• Quotient rule: a to power m divided by a to power n equals a to power m minus n
• Power rule: (a to power m) to power n equals a to power m times n
• Product to a power: (ab) to power m equals a to power m times b to power m
• Quotient to a power: (a/b) to power m equals a to power m divided by b to power m

Negative and Fractional Exponents

• Negative exponent: a to power negative n equals 1 divided by a to power n
• 2 to power negative 3 equals 1/8
• Fractional exponent: a to power 1/n is the nth root of a
• a to power m/n equals nth root of a to power m
• 2 to power 1/2 equals square root of 2 approximately 1.414

Scientific Notation

• Scientific notation expresses large and small numbers in compact form
• Form: a times 10 to power n where 1 is less than or equal to a and a is less than 10
• 500000 in scientific notation is 5 times 10 to power 5
• 0.0007 in scientific notation is 7 times 10 to power negative 4
• Used extensively in science, physics, and astronomy for very large or small quantities

Applications of Exponents

• Compound interest calculations use exponential growth formulas
• Population growth modeled using exponential functions
• Radioactive decay described using negative exponentials
• Computer science uses powers of 2 for binary and data storage units
• Technology follows Moore's Law, which is exponential progression

Key Terms

• Exponent: Number indicating how many times the base is multiplied by itself
• Base: Number that is repeatedly multiplied in exponential notation
• Power: Result of raising a base to an exponent
• Scientific Notation: Way of expressing numbers as product of a decimal and power of 10

What a negative exponent actually means

In Class 7 an exponent counted repeated multiplication: 2 to the power 3 means 2 multiplied by itself three times, giving 8. That reading breaks down the moment the exponent is negative — you cannot multiply a number by itself minus three times. So the meaning has to be extended, and it is extended in the one way that keeps the existing laws working.

Look at what happens going down a column: 2 cubed is 8, 2 squared is 4, 2 to the power 1 is 2. Each step down divides by 2. Continue the pattern and 2 to the power 0 must be 1, 2 to the power minus 1 must be one half, and 2 to the power minus 2 must be one quarter. That is the definition: a to the power minus m equals 1 over a to the power m, for any non-zero a.

The single most useful consequence is that a negative exponent flips a fraction. One half to the power minus 3 is 2 cubed, which is 8. Students lose marks by writing that it is minus 8 — a negative exponent never makes the answer negative. It takes the reciprocal, and nothing else.

Why standard form exists

Some quantities are absurdly large and some absurdly small, and both are painful to write and worse to compare. The diameter of a red blood cell and the distance to a star cannot sensibly share a page in ordinary decimal notation.

Standard form, also called scientific notation, writes any number as k times a power of 10, where k is at least 1 and less than 10. That constraint on k is not decoration — it is what makes the form unique, so two numbers can be compared by looking at their powers of 10 first and only then at k.

The laws of exponents — the complete list

These hold for any non-zero integer a and b and any integers m and n, including negative ones. That extension to negative exponents is exactly what this chapter adds to what you learnt in Class 7.

• Product with the same base: a to the m times a to the n equals a to the m plus n. Add the exponents.
• Quotient with the same base: a to the m divided by a to the n equals a to the m minus n. Subtract the exponents.
• Power of a power: a to the m, all raised to n, equals a to the m times n. Multiply the exponents.
• Same exponent, different bases: a to the m times b to the m equals a times b, all to the m.
• Quotient with the same exponent: a to the m divided by b to the m equals a over b, all to the m.
• Zero exponent: a to the power 0 equals 1 for every non-zero a. This follows from the quotient law, since a to the m divided by a to the m is both 1 and a to the power 0.
• Negative exponent: a to the minus m equals 1 over a to the m. Equivalently, a fraction raised to a negative power is the reciprocal fraction raised to the positive power.

Writing a number in standard form

The method is the same for large and small numbers; only the sign of the power changes.

• Place the decimal point immediately after the first non-zero digit. That number is k, and it will automatically satisfy 1 is less than or equal to k, which is less than 10.
• Count how many places the decimal point moved. That count is the magnitude of the power of 10.
• If the original number was large, so the point moved left, the power is positive. If the original number was small, so the point moved right, the power is negative.
• Check by sanity: a number bigger than 1 always gets a positive power, a number smaller than 1 always gets a negative power. If your sign disagrees with that, you have made an error.
• To convert back from standard form, move the decimal point the stated number of places — right for a positive power, left for a negative one — filling with zeros as needed.

Comparing and combining numbers in standard form

Standard form is not only a way of writing numbers; it is a way of comparing them quickly, and CBSE tests that.

• To compare two numbers in standard form, compare the powers of 10 first. The number with the larger power is larger, regardless of k, because k is confined to the range 1 to 10.
• Only if the powers are equal do you compare the values of k.
• To add or subtract numbers in standard form, first rewrite them so both have the same power of 10, then add or subtract the k values and keep the common power.
• After adding, the result may no longer be in standard form — if k has become 10 or more, adjust by increasing the power of 10 by one and dividing k by 10.

Solved Numericals — Step by Step

Worked Examples

• 2⁻³ is one eighth, not minus eight — The negative sign is in the exponent, not the base. It signals a reciprocal, so 2⁻³ = 1/2³ = 1/8. The answer stays positive.
• (2/5)⁻² becomes (5/2)² — A negative exponent on a fraction flips the fraction and makes the exponent positive, giving 25/4.
• 10⁵ ÷ 10⁵ shows why a⁰ = 1 — Any number divided by itself is 1, and the quotient law gives 10⁰. Both must be the same, so 10⁰ = 1.
• 43,000,000 in standard form — Put the point after the 4 to get 4.3, count 7 places moved left, and the number is greater than 1 so the power is positive: 4.3 × 10⁷.

Common Mistakes Students Make

• Treating a negative exponent as a negative answer. 3⁻² is 1/9, not −9. The minus sign moves the term to the denominator; it never changes the sign of the result.
• Multiplying the exponents when the bases are multiplied. For a^m × a^n you ADD the exponents; you multiply them only for a power of a power, (a^m)^n.
• Writing 0⁰ or applying a⁰ = 1 to a = 0. Every exponent law in this chapter is stated for a non-zero base, and 0⁰ is not defined.
• Leaving k outside the range 1 to 10 in standard form. Writing 0.56 × 10⁻⁵ or 34 × 10⁶ is not standard form; k must be at least 1 and less than 10.
• Getting the sign of the power backwards. A number smaller than 1 always takes a negative power; a number larger than 1 always takes a positive power.
• Adding numbers in standard form without first matching the powers of 10. You cannot add the k values until both terms carry the same power.

Where This Shows Up in Real Life

Reading measurements in science

Standard form is how every quantity in physics and chemistry is written once it leaves the everyday range. The charge on an electron, the mass of a proton, the wavelength of visible light and the distance to the nearest star are all quoted as k times a power of 10, and the questions in later classes assume you can move between that form and ordinary decimals without thinking about it. The rules in this chapter are therefore not a self-contained exercise — they are the notation the rest of school science is written in.

Comparing sizes across wildly different scales

Standard form turns an awkward comparison into a subtraction. A bacterium at about 10⁻⁶ m and a human at about 10⁰ m differ by six powers of 10, so the human is roughly a million times larger. Once both numbers are in standard form the comparison is read off the exponents, which is far quicker and far less error-prone than counting zeros.

Digital storage and powers of 2

File sizes and memory are built on powers of 2, so a kilobyte is 2¹⁰ bytes and a megabyte is 2²⁰. Working out how many files of one size fit into a disk of another is exactly the quotient law — subtract the exponents. The same reasoning explains why doubling a quantity repeatedly grows it so quickly.

Board-Style Questions with Model Answers

State the law of exponents used when dividing two powers of the same base, and use it to simplify 7⁻³ ÷ 7⁻⁵. (3 marks)

The law states that a to the power m divided by a to the power n equals a to the power m minus n, for any non-zero a. Applying it, 7⁻³ ÷ 7⁻⁵ equals 7 to the power minus 3 minus minus 5, which is 7 to the power minus 3 plus 5, that is 7². Therefore the value is 49. Note that subtracting a negative exponent increases the result, which is a frequent source of sign errors.

Explain why a⁰ = 1 for every non-zero a. (2 marks)

Consider a to the power m divided by a to the power m. Since any non-zero quantity divided by itself is 1, the value is 1. Applying the quotient law to the same expression gives a to the power m minus m, which is a to the power 0. Both expressions describe the same quantity, so a to the power 0 must equal 1. The base must be non-zero because division by zero is not defined.

Express 4.5 × 10⁻⁴ in ordinary decimal notation and state which is larger, 4.5 × 10⁻⁴ or 9.1 × 10⁻⁵. (3 marks)

For 4.5 × 10⁻⁴ the decimal point moves four places to the left, giving 0.00045. To compare the two numbers, compare the powers of 10 first: minus 4 is greater than minus 5, so 4.5 × 10⁻⁴ is the larger number. The value of k does not need to be considered here, because k is always at least 1 and less than 10, so a difference in the power of 10 always decides the comparison.

Simplify (2⁻¹ × 3⁻¹)⁻¹ ÷ 6⁻¹, showing each law used. (5 marks)

First simplify inside the bracket. Using the law that powers with the same exponent and different bases can be combined, 2⁻¹ × 3⁻¹ equals (2 × 3)⁻¹, that is 6⁻¹. The expression becomes (6⁻¹)⁻¹ ÷ 6⁻¹. Using the power of a power law, (6⁻¹)⁻¹ equals 6 to the power minus 1 times minus 1, which is 6¹, that is 6. Now divide using the quotient law: 6¹ ÷ 6⁻¹ equals 6 to the power 1 minus minus 1, which is 6², that is 36. The laws used are the same-exponent product law, the power of a power law, and the quotient law.

The thickness of a sheet of paper is 0.0016 cm. Express this in standard form and find the total thickness of a stack of 500 such sheets, giving the answer in standard form. (4 marks)

In standard form 0.0016 cm is 1.6 × 10⁻³ cm, since the first non-zero digit is 1, k is 1.6, and the decimal point moves three places to the right for a number smaller than 1. For 500 sheets the total thickness is 500 × 1.6 × 10⁻³ cm. Writing 500 as 5 × 10², the product is 5 × 1.6 × 10² × 10⁻³, which is 8 × 10⁻¹ cm. This equals 0.8 cm, which is a reasonable thickness for a stack of 500 sheets and so serves as a check on the working.

Assertion–Reason Questions (CBSE Format)

For each question choose: (a) both A and R are true and R is the correct explanation of A; (b) both are true but R is not the correct explanation; (c) A is true but R is false; (d) A is false but R is true.

Quick Self-Check

Key Terms Glossary

Exponent

The small raised number that tells how many times the base is used as a factor; also called the power or index.

Base

The number that is raised to a power. In 5³ the base is 5.

Negative exponent

An exponent less than zero; a to the minus m is defined as 1 divided by a to the m, for non-zero a.

Reciprocal

The multiplicative inverse of a number; the reciprocal of a is 1 over a, and a negative exponent produces it.

Zero exponent

Any non-zero number raised to the power 0 equals 1.

Standard form

Writing a number as k × 10ⁿ where k is at least 1 and less than 10 and n is an integer; also called scientific notation.

Product law

For the same non-zero base, a to the m times a to the n equals a to the m plus n.

Quotient law

For the same non-zero base, a to the m divided by a to the n equals a to the m minus n.

Power of a power

Raising a power to another power multiplies the exponents: (a to the m) to the n equals a to the mn.

Expanded form

Writing a number using powers of 10 for each place value, for example 4032 as 4 × 10³ + 0 × 10² + 3 × 10¹ + 2 × 10⁰.

Index

Another word for exponent; the plural is indices, and the exponent rules are often called the laws of indices.

One-Minute Revision

• A negative exponent means reciprocal, never a negative answer: 3⁻² = 1/9.
• A fraction to a negative power flips: (2/5)⁻² = (5/2)² = 25/4.
• Same base multiplied — add exponents. Same base divided — subtract. Power of a power — multiply.
• a⁰ = 1 for every non-zero a; 0⁰ is not defined.
• Standard form is k × 10ⁿ with k at least 1 and less than 10. That range is what makes it standard.
• Number bigger than 1 gives a positive power; number smaller than 1 gives a negative power. Check the sign this way every time.
• To compare numbers in standard form, compare the powers of 10 first, and only then compare k.
• Before adding or subtracting in standard form, make the powers of 10 equal.

Syllabus reference: NCERT Class 8 Mathematics — Exponents and Powers. Builds directly on the Class 7 chapter of the same name and is used again in Class 9 Number Systems.

Frequently Asked Questions

Why does any number to the power 0 equal 1?

Using the quotient rule: a to power n divided by a to power n equals a to power 0. Since a to power n divided by itself equals 1, a to power 0 equals 1.

What does a negative exponent mean?

A negative exponent means the reciprocal of the base raised to the positive exponent. For example, 2 to power negative 3 equals 1/(2 to power 3) which equals 1/8.

Why does a negative exponent not make the answer negative?

The minus sign sits on the exponent, not on the base, and the exponent controls how many times the base is used as a factor rather than the sign of the result. The definition a to the minus m equals 1 over a to the m produces a reciprocal, and taking the reciprocal of a positive number leaves it positive. So 2⁻³ is 1/8. A negative result only appears if the base itself is negative, as in (−2)³ = −8.

How do I decide whether the power of 10 is positive or negative in standard form?

Look at the size of the original number. If it is greater than or equal to 10 the decimal point moves left and the power is positive. If it is less than 1 the point moves right and the power is negative. A number already between 1 and 10 takes the power 0. Checking against this rule catches most sign errors instantly, because a tiny number can never have a positive power.

Is 34 × 10⁵ in standard form?

No. The value is correct as a number, but standard form additionally requires that k lies between 1 and 10, and 34 does not. Convert it by moving the decimal point one place left in k and compensating with one more power of 10, giving 3.4 × 10⁶. The constraint on k is what makes standard form unique for each number, which is what allows two numbers to be compared by their powers.

Do the laws of exponents still work when the exponents are negative?

Yes, and that is the main point of the chapter. Every law you learnt in Class 7 for positive exponents continues to hold for negative integers, provided the base is non-zero. For example the quotient law gives 7⁻³ ÷ 7⁻⁵ = 7⁻³⁻⁽⁻⁵⁾ = 7², so the laws are applied exactly as before — you simply have to handle the signs of the exponents carefully.

More Class 8 Mathematics Revision Notes

• Rational Numbers
• Squares and Square Roots
• Mensuration
• Linear Equations in One Variable
• Squares and Square Roots
• Cubes and Cube Roots

🤖 Stuck on any of these? Ask Syllab's free AI Tutor to explain step by step →