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This page sets out the method for fractional & negative 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 Fractional & negative indices topic page.
When you see a number raised to a fraction, especially a negative fraction, deal with each part of the index in turn.
When you see an equation like , write every number as a power of one small prime.
The bottom of the fraction power is the root. The top is the power you apply next.
A positive whole power just multiplies the base by itself.
A negative index means one over the positive power. 4^(-2) is 1 over 4², which is 1/16. It does not make the answer negative.
For 27^(-2/3), cube-root first, square, then take the reciprocal.
A power of 1/2 is the square root. 9^(1/2) is 3.
Do the root, then the power, then the reciprocal if the index is negative.
a^-n = 1/a^n. Start from a power you already know. 5^2 = 25, so 5^-2 = 1/25. As a decimal, 1/25 = 0.04. 2^-3 = 1/8 = 0.125. Nothing in that step multiplies by -1. A negative base is a different instruction, and it is not what a negative index means.
Apply the minus before you do anything else, or apply it at the end. Both orders work if you are careful. 16^-1 = 1/16. (1/5)^-2 = 5^2 = 25, because a negative index on a unit fraction flips it. Students who write -25 for 5^-2 have mixed the sign of the index with the sign of the value.
Zero still sits between the positive and negative powers. 5^0 = 1. Then 5^-1 = 1/5 and 5^1 = 5. The negative powers get smaller as the index becomes more negative, as long as the base is greater than 1. That pattern is a useful check: 5^-2 should be smaller than 5^-1.
Method: 1. A negative index means the reciprocal of the positive power. 2. Work out the positive power first. 3. Then write 1 over that result.
The index 1/n means the nth root. 8^1/3 = 2, because 2^3 = 8. 27^1/3 = 3. 16^1/4 = 2, because 2^4 = 16. 81^1/4 = 3. You are looking for the number which, raised to the denominator, gives the base. On a non-calculator paper the base has been chosen so the root is a small whole number.
A square root is the index 1/2. 9^1/2 = 3 and (4/9)^1/2 = 2/3. Take the root of the numerator and the root of the denominator. Do not square the fraction. The index 1/2 undoes a square. It does not apply another one.
If the root is not obvious, factor the base. 32^1/5: 32 = 2^5, so the fifth root is 2. That prime-power reading is the same skill you will use for equations later in the lesson. Write 32^1/5 = (2^5)^1/5 = 2^1 = 2.
Method: 1. The denominator of the index tells you which root to take. 2. Find the number which, raised to that denominator, gives the base. 3. Check by raising your root back to the denominator.
The index m/n means 'nth root, then raise to the power m'. You may power first and then take the root. Root first keeps the numbers smaller. 8^2/3: cube root of 8 is 2, then 2^2 = 4. Doing the square first gives 8^2 = 64, and the cube root of 64 is also 4. Same answer, heavier arithmetic.
27^2/3: cube root of 27 is 3, then 3^2 = 9. 16^3/4: fourth root of 16 is 2, then 2^3 = 8. 81^3/4: fourth root of 81 is 3, then 3^3 = 27. Write the root on one line and the power on the next so a marker can see both parts of the fraction.
A negative fractional index does both jobs. 16^-3/4 = 1/16^3/4. From the line above, 16^3/4 = 8, so 16^-3/4 = 1/8 = 0.125. Find the positive power in full, then take the reciprocal. Trying to 'make the root negative' is not a third operation. There is only the reciprocal and the positive fractional power.
Method: 1. Take the root named by the denominator first. 2. Then raise that root to the numerator. 3. Root first keeps the numbers smaller than powering first.
A fractional base with a negative index should be flipped first. (4/9)^-1/2 = (9/4)^1/2. The square root of 9/4 is 3/2, which is 1.5. Leaving the fraction as 4/9 and then trying to attach the minus to 2/3 produces -2/3, which is the wrong sign and the wrong way up. Flip, then root.
(8/27)^-2/3 flips to (27/8)^2/3. The cube root of 27/8 is 3/2, and (3/2)^2 = 9/4 = 2.25. Again the negative index was used up by the flip. The remaining index 2/3 is positive. Check that a negative power of a number between 0 and 1 comes out greater than 1, because you have taken a reciprocal of a small positive number.
For 9^x = 27, write both sides as powers of 3. 9 = 3^2 and 27 = 3^3, so (3^2)^x = 3^3. That is 3^2x = 3^3. The bases match, so the indices match: 2x = 3 and x = 3/2 = 1.5. Check by evaluating 9^3/2. The square root of 9 is 3, and 3^3 = 27. The equation holds.
Method: 1. A negative index on a fraction flips the fraction first. 2. Then apply the root or power that remains. 3. If two powers are equal, match the bases and compare the indices.
Fractional and negative index questions are set without a calculator, so the numbers are always perfect powers. Learn the squares up to 15^2, the cubes 1, 8, 27, 64, 125, 1000, and the fourth and fifth powers that turn up in questions: 2^4 = 16, 3^4 = 81, 2^5 = 32 and 10^4 = 10 000. When you see 81^3/4, 32^-3/5 or 64^2/3, you should recognise 81 as 3^4, 32 as 2^5 and 64 as 4^3 or 2^6.
If a number is not obvious, use a factor tree or repeated division by small primes to find its base. A number such as 64 = 2^6 can give a square root of 8, a cube root of 4 and a sixth root of 2. The denominator of the index tells you which root to take, so you need only the one root that gives a whole number. If no whole-number root exists, check that you have copied the question correctly.
Method: 1. Read the denominator of the index to decide which root to take. 2. Find the root first, because it keeps the numbers small. 3. Apply the numerator of the index as a power to that root. 4. If the index is negative, take the reciprocal at the start or the end, but do it exactly once.
Tip: Roots first, then powers: 16^3/2 is 4^3 = 64, not √(4096).
When the base is a fraction, a negative index flips the fraction. So (2/3)^-3 = (3/2)^3. Do the flip as the very first line, because it removes the negative sign and leaves an ordinary calculation. Then apply the root and the power to the top and the bottom separately. For example (27/8)^2/3 = (3/2)^2 = 9/4, because the cube root of 27 is 3 and the cube root of 8 is 2.
Check how the question wants the answer written: as an improper fraction, as a mixed number, or as a fraction in its simplest form. The fraction 27/8 is correct as an improper fraction and 33/8 is the same value as a mixed number. Do not turn the answer into a decimal unless asked. A common slip is to flip the fraction but forget the power, or to flip a whole number: 2^-3 is 1/8, not 8.
Method: 1. Flip the fraction and remove the minus sign from the index. 2. Take the root of the numerator and of the denominator. 3. Raise both to the power in the numerator of the index. 4. Give the answer in the form asked for, such as a mixed number or a simplest-form fraction.
To solve 4^x = 1/32, write both sides as powers of the same base. Here 4 = 2^2 and 1/32 = 2^-5, so 2^2x = 2^-5. Equal powers of the same base must have equal indices, so 2x = -5 and x = -5/2. Notice that the answer can be negative and can be a fraction. These questions are marked on the change of base, the equation for the indices and the final value, so show each step.
A second kind is "Given that 27^-1/3 = 81^1/4 ÷ 3^x, find x." Work out each side that is only numbers first: 27^-1/3 = 1/3 and 81^1/4 = 3. Then 1/3 = 3 ÷ 3^x, so 3^x = 9 and x = 2. Always check by putting your value of x back into the original equation and making sure both sides agree.
Method: 1. Pick a common base, usually the smaller prime such as 2 or 3. 2. Rewrite each side as a power of that base. 3. Equate the indices and solve the resulting equation. 4. Substitute back to check both sides match.
Algebra questions ask you to simplify an expression such as (27x^6)^2/3. Apply the outer power to every factor. The number gives 27^2/3 = 9 and the letter gives x^6 × 2/3 = x^4, so the result is 9x^4. Multiply the indices when a power is raised to a power, add them when you multiply the same letter, and subtract them when you divide.
Take care with negative indices on letters. A term such as y^-2 in the denominator moves to the numerator, so dfrac2y^2y^-2 = 2y^4, because 2 - (-2) = 4. Write each step so that a sign slip costs one mark rather than three. When you are unsure, test your answer by substituting a small number such as x = 2 into both the original expression and your simplified form and check they agree.
Method: 1. Apply the outer power to the number and to each letter separately. 2. Multiply indices for a power of a power. 3. Add indices when multiplying and subtract when dividing, writing negatives in brackets. 4. Test the answer by substituting a small value.
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