MathsCBSE Class 8 · Power Play

Laws of Exponents, Zero and Negative Powers for Class 8

Updated 17 October 2026 · Fact-checked

The laws of exponents are: a^m x a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn), a^m x b^m = (ab)^m, with a^0 = 1 and a^(-n) = 1/a^n for a not equal to 0. School exams ask for simplification to a single power, solving for an unknown exponent, comparing large powers by a common exponent, and unit digits of powers.

Each step down halves the value and lowers the exponent by 1, so 2^0 = 1 and 2^-1 = 1/2: a negative power is a reciprocal, not a negative number.

Illustration drawn for QuestionBuddy.

Understand Laws of Exponents, Zero and Negative Powers

An exponential form $a^{n}$ means $a$ multiplied by itself $n$ times: $5^{4} = 5 \times 5 \times 5 \times 5 = 625$ . Here 5 is the base and 4 is the exponent (power). Do not confuse $4 \times 3 = 12$ with $4^{3} = 64$ . Note also that $(- 4)^{3} = - 64$ , while an even exponent makes a negative base positive: $(- 2)^{4} = 16$ .

Why the laws work. Count the factors.

*Product law.* $a^{m} \times a^{n} = a^{m + n}$ because you are joining $m$ copies and $n$ copies of $a$ . For example $3^{4} \times 3^{3} = 3^{7} = 2187$ .
*Power of a power.* $(a^{m})^{n} = a^{mn}$ , since $n$ groups of $m$ factors make $mn$ factors. Also $(a^{m})^{n} = (a^{n})^{m}$ , which is why $2^{10}$ can be $(2^{5})^{2}$ or $(2^{2})^{5}$ .
*Same exponent, different bases.* $a^{m} \times b^{m} = (ab)^{m}$ , so $2^{5} \times 5^{5} = 1 0^{5}$ .
*Quotient law.* $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ for $a \neq = 0$ , because common factors cancel.

Zero exponent. Take $m = n$ in the quotient law: $\frac{a ^{n}}{a ^{n}} = a^{0}$ , but the left side is 1. So $a^{0} = 1$ for every $a \neq = 0$ . The base cannot be 0 because the quotient would involve dividing by zero.

Negative exponents. Take $m < n$ : halving a line of length $2^{4}$ five times gives $2^{4 - 5} = 2^{- 1} = \frac{1}{2}$ . In general $a^{- n} = \frac{1}{a ^{n}}$ and $\frac{1}{a ^{- n}} = a^{n}$ . The sign of the exponent tells you whether the power sits on the top or the bottom of a fraction; it does not make the number negative. For example $(- 5)^{- 3} = - \frac{1}{125}$ , and $(- 3)^{- 2} = \frac{1}{9}$ .

All the laws continue to hold when the exponents are any integers, positive, zero or negative.

Power lines. Arranging the powers of 4 in a line, $\dots, 4^{- 2}, 4^{- 1}, 4^{0}, 4^{1}, 4^{2}, \dots$ , each step to the right multiplies by 4 and each step to the left divides by 4.

Comparing large powers. To compare $a^{m}$ and $b^{n}$ , rewrite both with a common exponent (or a common base). For instance $3^{40} = 9^{20}$ and $2^{60} = 8^{20}$ , so $3^{40}$ is larger.

Unit digits. The unit digits of powers repeat in a cycle. For 3 the cycle is 3, 9, 7, 1, and for 7 it is 7, 9, 3, 1. Divide the exponent by 4 and use the remainder.

This page covers the laws; large numbers and scientific notation are on the next topic page of this chapter.

Laws of Exponents, Zero and Negative Powers: formulas to remember

Product of powers (same base): $a^{m} \times a^{n} = a^{m + n}$; Add the exponents.
Quotient of powers (same base): $\frac{a ^{m}}{a ^{n}} = a^{m - n}, a \neq = 0$; Subtract the exponents.
Power of a power: $(a^{m})^{n} = a^{mn} = (a^{n})^{m}$; Multiply the exponents.
Same exponent: $a^{m} \times b^{m} = (ab)^{m}, \frac{a ^{m}}{b ^{m}} = (\frac{a}{b})^{m}$; Used to join different bases.
Zero exponent: $a^{0} = 1, a \neq = 0$; From a^n divided by a^n.
Negative exponent: $a^{- n} = \frac{1}{a ^{n}}, \frac{1}{a ^{- n}} = a^{n}$; A reciprocal, not a negative number.

How to solve Laws of Exponents, Zero and Negative Powers questions

Simplify and compare powers in a fixed order so you do not miss a step.

1Break every base into primes (for example 12 = 2^2 x 3, 9 = 3^2, 8 = 2^3).
2Apply the power-of-a-power law to every bracket first.
3Collect each base: add exponents in products and subtract in quotients, remembering that subtracting a negative adds.
4Convert any remaining negative exponent to a reciprocal only at the end.
5To solve for an unknown exponent, make the bases the same and equate the exponents.
6To compare two powers, bring them to the same exponent or the same base.
7Check with one small substitution or a quick numerical value.

Laws of Exponents, Zero and Negative Powers in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Exponential growth by folding

Suppose a sheet of paper is 0.001 cm thick (the value used in the chapter) and it could be folded in half again and again, doubling the stack each time.

Find the thickness after 10 folds and after 20 folds.

Try it first, then show the solutionSolution

After $n$ folds the thickness is $0.001 \times 2^{n}$ cm.
10 folds: $0.001 \times 1024 = 1.024$ cm.
20 folds: $2^{20} = 1048576$ , so $0.001 \times 1048576 = 1048.576$ cm, about 10.5 m.

Answer: 1.024 cm after 10 folds; 1048.576 cm (about 10.5 m) after 20 folds.

Memory hook: Ten more folds multiplies the thickness by 1024, not by adding a fixed amount.

Counting codes

Imagine a locker that opens with a 5-digit code using the digits 0 to 9 in every place.

How many different codes are possible, and how many are there if one more digit is added?

Try it first, then show the solutionSolution

Each of the 5 places has 10 choices: $10 \times 10 \times 10 \times 10 \times 10 = 1 0^{5} = 1, 00, 000$ .
One more digit multiplies by 10: $1 0^{6} = 10, 00, 000$ .

Answer: 1,00,000 codes; 10,00,000 codes with six digits.

Memory hook: Each new independent choice multiplies, so the count is a power.

Tricks to remember Laws of Exponents, Zero and Negative Powers

Short memory aids. Say each one aloud once, then use it on a question the same day.

Product adds, power multiplies, quotient subtracts
Same base: multiplying powers means adding exponents, a power of a power means multiplying exponents, and dividing means subtracting. Say it as add, times, minus.
Use it when: Whenever you meet a simplification with brackets.
Negative exponent flips the floor
A negative exponent sends the power to the other floor of the fraction. 2^(-3) is the same as 1 over 2^3.
Use it when: Turning negative powers into fractions and back.
Anything to the power zero is one
Any non-zero number to the power 0 is 1, because the power has divided itself out.
Use it when: Quick evaluation of expressions with zero exponents.
Same exponent, then compare
To compare powers with different bases and exponents, make the exponents equal, then compare the bases.
Use it when: Questions such as 3 to the 40 versus 2 to the 60.

Laws of Exponents, Zero and Negative Powers: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Reciprocal powers · 1 mark · MCQBoard level

Which of the following equals $\frac{1}{64}$ ? (a) $2^{- 5}$ (b) $4^{- 3}$ (c) $8^{- 3}$ (d) $(- 2)^{- 5}$

Try it first, then show the solutionSolution

$4^{- 3} = \frac{1}{4 ^{3}} = \frac{1}{64}$ .
Others: $2^{- 5} = \frac{1}{32}$ , $8^{- 3} = \frac{1}{512}$ , $(- 2)^{- 5} = - \frac{1}{32}$ .

Answer: (b) 4^(-3)

Example 2: Simplify to a single power · 2 marks · Short answerBoard level

Simplify $\frac{7 ^{- 3} \times 7 ^{8}}{7 ^{2}}$ and write it as a power of 7 and as a number.

Try it first, then show the solutionSolution

Numerator: $7^{- 3 + 8} = 7^{5}$ . (1 mark)
Quotient: $7^{5 - 2} = 7^{3} = 343$ . (1 mark)

Answer: 7^3 = 343

Example 3: Brackets, negative powers and a reciprocal · 3 marks · Short answerHarder board level

Evaluate $(2^{- 1} + 4^{- 1})^{- 2} \div (3^{- 2})^{- 1}$ .

Try it first, then show the solutionSolution

$2^{- 1} + 4^{- 1} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}$ , and $(\frac{3}{4})^{- 2} = \frac{16}{9}$ . (1 mark)
$(3^{- 2})^{- 1} = 3^{2} = 9$ . (1 mark)
$\frac{16}{9} \div 9 = \frac{16}{81}$ . (1 mark)

Answer: 16/81

Example 4: Solve for an exponent · 3 marks · Short answerHarder board level

Find $x$ if $2^{x + 3} \times 4^{x - 1} = 1024$ .

Try it first, then show the solutionSolution

Write $4^{x - 1} = (2^{2})^{x - 1} = 2^{2 x - 2}$ . (1 mark)
Left side $= 2^{x + 3 + 2 x - 2} = 2^{3 x + 1}$ , and $1024 = 2^{10}$ . (1 mark)
Equate exponents: $3 x + 1 = 10$ , so $x = 3$ . Check: $2^{6} \times 4^{2} = 64 \times 16 = 1024$ . (1 mark)

Answer: x = 3

Example 5: Ordering large powers · 4 marks · Long answerHarder board level

Arrange $2^{60}$ , $3^{40}$ and $5^{20}$ in ascending order, without evaluating them.

Try it first, then show the solutionSolution

Bring all to exponent 20: $2^{60} = (2^{3})^{20} = 8^{20}$ . (1 mark)
$3^{40} = (3^{2})^{20} = 9^{20}$ . (1 mark)
$5^{20}$ is already $5^{20}$ . (1 mark)
With the same exponent, the larger base gives the larger power: $5 < 8 < 9$ , so $5^{20} < 2^{60} < 3^{40}$ . (1 mark)

Answer: 5^20 < 2^60 < 3^40

Where students lose marks in Laws of Exponents, Zero and Negative Powers

The slips examiners see most often, and what to write instead.

Mistake: Multiplying the exponents in a product of powers
The product law adds exponents; only a power of a power multiplies.
Wrong working
" $3^{4} \times 3^{3} = 3^{12}$ "
Right working
$3^{4} \times 3^{3} = 3^{4 + 3} = 3^{7} = 2187$ .
Do this: Ask yourself: am I joining factors (add) or repeating a group (multiply)?
Usually costs: The entire part.
Mistake: Treating a negative exponent as a negative number
The exponent -n gives a reciprocal, not a negative value.
Wrong working
" $2^{- 3} = - 8$ "
Right working
$2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$ .
Do this: Flip the floor; keep the sign of the base separate.
Usually costs: 1 to 2 marks.
Mistake: Adding the bases in a product of powers
The laws apply to exponents of the same base, not to bases.
Wrong working
" $2^{3} \times 3^{3} = 5^{6}$ "
Right working
$2^{3} \times 3^{3} = (2 \times 3)^{3} = 6^{3} = 216$ .
Do this: Same base: add exponents. Same exponent: multiply the bases.
Usually costs: The part.
Mistake: Dropping the sign when subtracting a negative exponent
Subtracting -2 adds 2.
Wrong working
" $\frac{5 ^{3}}{5 ^{- 2}} = 5^{3 - 2} = 5$ "
Right working
$\frac{5 ^{3}}{5 ^{- 2}} = 5^{3 - (- 2)} = 5^{5} = 3125$ .
Do this: Put the second exponent in brackets before subtracting.
Usually costs: 1 mark.

Exam tips

Convert every number to prime powers first. Most simplifications then become exponent arithmetic.
In "solve for x" questions, make the bases equal before equating exponents; show that step because it earns its own mark.
Write the law you use at the side of each line. Examiners give method marks for it.
Leave answers as a single power only when asked; otherwise also give the number if it is small.
For comparisons, show the common exponent clearly, then compare the bases.

Practice questions on Laws of Exponents, Zero and Negative Powers

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): (-3)^(-2) = -1/9.
Reason (R): a^(-n) = 1/a^n for a not equal to 0.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Apply R to a base of -3 and an exponent of 2.
Answer
(d). $(- 3)^{- 2} = \frac{1}{( - 3 ) ^{2}} = \frac{1}{9}$ , which is positive, so A is false. R is the standard rule and is true.
Q2 · 2 marks · Very short answerBoard level
Write $\frac{1}{81}$ as a power of 3 and as a power of 9, each with a negative exponent.
Hint
$81 = 3^{4} = 9^{2}$ .
Answer
$\frac{1}{81} = 3^{- 4}$ and $\frac{1}{81} = 9^{- 2}$ .
Q3 · 3 marks · Short answerHarder board level
Simplify $\frac{1 2 ^{4} \times 9 ^{3} \times 4}{6 ^{3} \times 8 ^{2} \times 27}$ and give the value.
Hint
Convert every base to powers of 2 and 3.
Answer
$1 2^{4} = 2^{8} \times 3^{4}$ , $9^{3} = 3^{6}$ , $4 = 2^{2}$ : numerator $= 2^{10} \times 3^{10}$ . $6^{3} = 2^{3} \times 3^{3}$ , $8^{2} = 2^{6}$ , $27 = 3^{3}$ : denominator $= 2^{9} \times 3^{6}$ . Quotient $= 2^{1} \times 3^{4} = 2 \times 81 = 162$ .
Q4 · 4 marks · Case-basedHOTS / competency
Case study: suppose a sheet of paper is 0.001 cm thick and is folded in half repeatedly, doubling its thickness at each fold (an idealised situation).
(i) Find its thickness after 10 folds. (1 mark)
(ii) Find its thickness after 20 folds in metres, correct to one decimal place. (1 mark)
(iii) What is the least number of folds that makes the stack taller than 1 metre? Show that one fewer fold is not enough. (2 marks)
Hint
Thickness after n folds is 0.001 x 2^n cm. 1 m = 100 cm, so you need 2^n > 100000.
Answer
(i) $0.001 \times 2^{10} = 0.001 \times 1024 = 1.024$ cm. (ii) $0.001 \times 2^{20} = 0.001 \times 1048576 = 1048.576$ cm $\approx 10.5$ m. (iii) Need $0.001 \times 2^{n} > 100$ , i.e. $2^{n} > 100000$ . $2^{16} = 65536 < 100000$ (thickness 65.536 cm) and $2^{17} = 131072 > 100000$ (131.072 cm). So the least number is 17.
Q5 · 4 marks · Long answerHOTS / competency
Find the unit digit of $3^{2025} + 7^{2025}$ . Hence say what you can conclude about its divisibility by 10.
Hint
Unit digits of powers of 3 and 7 cycle with period 4. 2025 = 4 x 506 + 1.
Answer
Powers of 3 end in 3, 9, 7, 1 repeating with period 4; powers of 7 end in 7, 9, 3, 1. Since $2025 = 4 \times 506 + 1$ , the exponent leaves remainder 1, so $3^{2025}$ ends in 3 and $7^{2025}$ ends in 7. The sum ends in $3 + 7 = 10$ , so its unit digit is 0. A natural number ending in 0 is divisible by 10, so $3^{2025} + 7^{2025}$ is divisible by 10.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is a^0 for a not equal to 0?Show answer
1.
2.Simplify a^m x a^n.Show answer
a^(m+n).
3.Simplify (a^m)^n.Show answer
a^(mn).
4.Write 2^(-3) as a fraction.Show answer
1/8.
5.What is the sign of (-2)^4 and (-2)^5?Show answer
Positive (16) and negative (-32).
6.What is the cycle of unit digits of powers of 3?Show answer
3, 9, 7, 1 repeating.

Laws of Exponents, Zero and Negative Powers: frequently asked questions

Why is a to the power 0 equal to 1?

Take the quotient law a^n divided by a^n. The numerator and denominator are equal, so the quotient is 1. The law gives a^(n-n) = a^0, so a^0 must be 1. The base must not be 0.

Can the exponent be any integer?

Yes. The chapter extends the laws from counting-number exponents to all integers, including 0 and negatives, and they still hold.

How do I compare 4 cubed and 3 to the 4?

Evaluate small cases: 4 cubed is 64 and 3 to the 4 is 81, so 3 to the 4 is greater. For large cases use a common exponent.

Is 0 to the power 0 defined?

The rule a^0 = 1 is stated for a not equal to 0 in this chapter, so do not use it for base 0.