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This page sets out the method for indices in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Indices topic page.
When you see the same letter multiplied or divided with different powers, sort the numbers first, then the powers.
When a whole bracket is raised to a power, every part inside gets that power.
2⁴ = 2 × 2 × 2 × 2 = 16.
The base is 2. The power is 4.
a^m × a^n = a^(m+n). The base stays.
2³ × 2² = 2⁵ = 32. Do not multiply the powers.
a^m ÷ a^n = a^(m−n).
2⁵ ÷ 2² = 2³ = 8.
(a^m)^n = a^(mn).
(2³)² = 2⁶ = 64.
a¹ = a. a⁰ = 1 for a ≠ 0.
2⁰ is 1, not 0.
2³ × 3² cannot be merged into one power.
Work out each power, then multiply.
Write the factors out when the powers are small, so the law has a reason. 4^3 × 4^2 = (4 × 4 × 4) × (4 × 4). That is five factors of 4, so it is 4^5. The indices 3 and 2 were added because the factor lists were joined. Adding the bases, or multiplying the bases, would describe a different sum.
The value is 4^5 = 1024. You can build it in steps: 4^2 = 16, 4^3 = 64, 4^4 = 256, 4^5 = 1024. Multiplying the separate powers also works: 4^3 = 64 and 4^2 = 16, and 64 × 16 = 1024. Use the law to get the single power, then evaluate if the question asks for a number.
2^5 × 2^3 = 2^8 = 256. Check the arithmetic of the index: 5 + 3 = 8, not 15. A common slip is to multiply the indices when the sign between the powers is a multiply. Multiplying indices belongs to a power of a power, which has brackets: (2^5)^3.
Method: 1. You can multiply powers only when the base is the same. 2. Add the indices and keep the base. 3. Check a small case by writing the factors out.
Division cancels matching factors. 3^4 ÷ 3^2 = (3 × 3 × 3 × 3) ÷ (3 × 3). Two factors cancel, and two factors of 3 remain, so the result is 3^2 = 9. The index law is a^m ÷ a^n = a^m-n. Here 4 - 2 = 2. As a check, 3^4 = 81 and 81 ÷ 9 = 9.
A bracket with an index outside means the whole power is repeated. (2^3)^2 = 2^3 × 2^3 = 2^6 = 64. The indices are multiplied: 3 × 2 = 6. Without the bracket, 2^3^2 means something different: the top power is worked out first, so it is 2^9. Follow the brackets that are printed.
If a coefficient sits outside, deal with the base first. 4 × 2^3 × 2^2 = 4 × 2^5 = 4 × 32 = 128. The 4 is not a power of 2 in this line, so it is not absorbed into the index. Only combine 2^3 and 2^2. Then multiply by the coefficient.
Method: 1. To divide powers of the same base, subtract the indices. 2. A power of a power means you multiply the indices. 3. Keep the base unchanged in both laws.
Dividing a power by itself leaves 1. 5^4 ÷ 5^4 = 5^0, and the value of that division is 1. So 5^0 = 1. The same idea gives a^0 = 1 for any non-zero a. The base is not turned into 0. 7^0 is 1, not 0 and not 7.
A negative index means a reciprocal. 2^-3 = 1/2^3 = 1/8 = 0.125. The minus sign does not make the value negative. It sends the power downstairs. 2^-4 = 1/16 = 0.0625. On a non-calculator paper, leave 1/16 if the question allows a fraction, or write the decimal if it asks for one.
You can mix the laws. (3^2)^3 ÷ 3^4 = 3^6 ÷ 3^4 = 3^2 = 9. Do the power of a power first, then the division. A zero index can appear in the middle of a line and be replaced by 1 at once: 6^5 ÷ 6^5 = 6^0 = 1. Keep the base the same all the way through.
Method: 1. Any non-zero number to the power 0 equals 1. 2. A negative index is the reciprocal of the positive power. 3. Work out the positive power, then write 1 over it.
2^3 × 5^2 looks tempting to combine, because 2 × 5 = 10. The index law does not allow that. The law adds indices only when the bases are already equal. Here one power is a product of twos and the other is a product of fives. Work them separately: 2^3 = 8 and 5^2 = 25. Then 8 × 25 = 200.
10^5 = 100000, which is nowhere near 200. Writing 10^5 multiplied the bases and then did something loose with the indices. Even 10^3 or 10^2 would be a guess. If you want a power of 10, you need the factors to be tens. 2^3 × 5^3 = (2 × 5)^3 = 10^3, and that works only because the indices match as well as the product of the bases.
Before you touch an index, ask one question: are the bases the same? If they are, add indices to multiply, subtract indices to divide, and multiply indices for a bracketed power. If they are not, evaluate each power or simplify each base on its own. Coefficients wait until that is done. Then do any addition or subtraction that BIDMAS still has waiting.
Method: 1. Do not add indices when the bases are different. 2. Work out each power on its own. 3. Then multiply those values if the question asks for a product.
Exam questions seldom test one law at a time. A typical question is "Simplify (3^-4 × 3^7)^3 ÷ 3^5. Give your answer as a power of 3." Work from the inside out. Inside the bracket, multiply the powers by adding the indices: -4 + 7 = 3, so you have (3^3)^3. Then a power of a power multiplies the indices: 3^9. Finally divide by subtracting: 9 - 5 = 4.
Write one line for each law so the marks are easy to find. Do not try to do the whole question in your head, because the negative index in the first bracket is a trap. The final answer must be in the form the question asks for. If it says "as a power of 3", write 3^4. If it says "work out the value", write 81. Do both if you are unsure; the power form shows your method.
Method: 1. Work out what is inside any brackets first, using the multiply or divide law. 2. Apply the power outside the bracket by multiplying the indices. 3. Combine the remaining terms by adding indices for a product or subtracting for a quotient. 4. Write the answer in the form requested, as a power or as a number.
Tip: The three index laws only work when the bases are the same. If they are not, change them to the same base first.
If the bases differ, rewrite them as powers of a common small number. 8 = 2^3, 16 = 2^4, 32 = 2^5 and 4 = 2^2 are all powers of 2. 9 = 3^2, 27 = 3^3 and 81 = 3^4 are all powers of 3. So 81 × 3^2 becomes 3^4 × 3^2 = 3^6, and 4 × 2^5 × 2^3 becomes 2^2 × 2^5 × 2^3 = 2^10.
Learn the small powers of 2 and 3 and of 5 and 10 by heart. They appear on every non-calculator paper. When a bracket contains a number like 4 raised to a power, write 4^n = (2^2)^n = 2^2n. Check numerically with a small value of n if you are not sure: put n = 2 in both the original and your answer, and they must match.
Method: 1. List the bases in the question and find a common small base. 2. Rewrite each number as a power of that base. 3. Use the index laws to combine the powers. 4. Check with a simple value, or work out both sides if they are numbers.
A negative index does not make the answer negative. It means "one over": 5^-2 = dfrac15^2 = 1/25 and 10^-3 = 1/1000 = 0.001. When the base is a fraction, a negative index flips it: (2/3)^-2 = (3/2)^2 = 9/4. Flip first, then apply the positive power to the top and bottom.
Questions often combine these values in a sum or difference, for example 4^-2 + 2^-1 = 1/16 + 1/2 = 9/16. Change each term to a fraction, find a common denominator, then add. Remember that any non-zero number to the power 0 is 1, and a number to the power 1 is itself. For example 6^0 + 6^1 + 6^2 = 1 + 6 + 36 = 43.
Method: 1. Replace each negative index by a reciprocal. 2. Flip a fractional base and make the index positive. 3. Work out each value as a fraction or a whole number. 4. Add or subtract using a common denominator, then simplify.
Sometimes the index is the unknown. "Find n if 2^n = 32" asks which power of 2 gives 32. Count up: 2, 4, 8, 16, 32, so n = 5. For 3^n+1 = 81, write 81 = 3^4, then match the indices: n + 1 = 4, so n = 3. Matching works only when the bases on both sides are the same.
Harder questions have different bases on each side, such as 4^x+1 = 2^3x-1. Write 4 = 2^2, so the left side is 2^2x+2. Now the bases agree, so set the indices equal: 2x + 2 = 3x - 1, which gives x = 3. Check by substituting back: both sides equal 256. The check takes ten seconds and catches sign slips.
Method: 1. Write both sides with the same base. 2. Equate the indices and form an equation. 3. Solve the equation for the unknown. 4. Substitute your value back to check both sides match.
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