Indices (powers), negative and fractional too
Learn how to multiply and divide powers using the multiplying machine. Worked examples, exam-style questions with answers.
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Download the PDF (658 KB)Why is 50 equal to 1, and what could 8-1/3 possibly mean?
Powers look like a list of rules to memorise, but they all come from one picture: a machine that starts at 1 and multiplies by the base each time you press its button. Once you see the machine, zero, negative and fractional powers make sense.
The picture to keep in your head: The multiplying machine
A power is a machine that starts at 1 and multiplies by the base again and again: 23 means start at 1 and press 'times 2' three times, so 1 → 2 → 4 → 8. A power of 0 means you never press the button, so you're still at 1. A negative power means running the machine backwards, dividing instead: 2-3 = 1/8. A fractional power is a fraction of the job: 81/3 asks which number, pressed three times, makes 8; that's the cube root, 2.
The method, every time
- Start at 1 and press the button as many times as the power says.
- If you multiply powers with the same base, add the number of presses.
- If you divide powers with the same base, subtract the number of presses.
- A zero power means you never press the button, so you stay at 1.
- A negative power means you run the machine backwards, dividing instead of multiplying.
Words you'll see
- base
- The number you multiply repeatedly in a power, like the 2 in 23.
- power
- How many times you press the multiplying machine button, like the 3 in 23.
- exponent
- Another name for the power, like the 3 in 23.
- negative
- Means you run the multiplying machine backwards, dividing instead of multiplying.
- fractional
- Means you do part of the job, like finding the cube root of 8 for 81/3.
Watch it explained
- Why any number to the power 0 is 1 (one-minute video)
Multiply: add the powers. Divide: subtract them.
The multiplying machine helps you understand powers. When you multiply numbers with the same base, you're just adding more presses. When you divide, you're undoing some presses. This means you add powers when multiplying and subtract them when dividing. The rule is: same base, add or subtract powers.
Watch out
- Multiplying the powers when multiplying numbers. Add the powers, not multiply them
- Using the laws on different bases. Only apply the rules when the base is the same
Zero means no presses; negative means divide
The multiplying machine starts at 1 and multiplies by the base each time. If you press the button 0 times, you’re still at 1. A negative power means you run the machine backwards, dividing instead. So 20 = 1 and 2-1 = 1/2. The rule is: a0 = 1 and a^-n = 1/an. A negative power never makes the answer negative.
Watch out
- Writing 4-2 = -16. Negative powers mean dividing, not making the answer negative
- Thinking anything to the power 0 is 0. Any number to the power 0 is 1, not 0
The bottom of the fraction is a root
81/3 asks: which number, put through the machine three times, makes 8? That's the cube root, 2. So in a fractional power the bottom number is a root and the top number is a power. Do the root first to keep the numbers small: 272/3 is the cube root of 27, squared.
Watch out
- Dividing by the bottom number. Take the root first, not the power
- Doing the power first and getting huge numbers. Always take the root before the power
Same laws, with letters
The multiplying machine works with letters too. When you square (2x³), you multiply the number and the power. Powers of the same letter add when you multiply. Let’s simplify (2x³)² × 3x⁻⁴.
Watch out
- Forgetting to square the number in (2x³)². Always square both the number and the power.
- Treating 3x⁻⁴ as (3x)⁻⁴. Only the x is raised to the power, not the 3.
23 exam-style indices (powers) questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · The laws
- 1Simplify 23 × 24.[2 marks]
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27
Multiply powers with same base: add exponents. 3 + 4 = 7. Answer: 27. - 2Simplify 58 ÷ 52.[2 marks]
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56
Divide powers with same base: subtract exponents. 8 − 2 = 6. Answer: 56. - 3Simplify (32)3.[2 marks]
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36
Power of a power: multiply exponents. 2 × 3 = 6. Answer: 36. - 4Simplify x5 × x2 ÷ x3.[2 marks]
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x4
Multiply and divide powers with same base: add and subtract exponents. 5 + 2 − 3 = 4. Answer: x4. - 5Simplify 42 × 43 ÷ (41)2.[2 marks]
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43
Multiply and divide powers with same base: add and subtract exponents. 2 + 3 − 2 = 3. Answer: 43.
B · Zero and negative
- 6Evaluate 70.[2 marks]
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1
70 means the multiplying machine never starts. So 1. Answer is 1. - 7Evaluate 5-1.[2 marks]
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0.2
5-1 means running the machine backwards once. So 1 ÷ 5 = 0.2. Answer is 0.2. - 8Evaluate 2-4.[2 marks]
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0.0625
2-4 means running the machine backwards four times. So 1 ÷ (2 × 2 × 2 × 2) = 1 ÷ 16 = 0.0625. Answer is 0.0625. - 9Evaluate (1/3)-2.[2 marks]
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9
A negative power flips the fraction: (1/3)-2 = 32 = 9. - 10Which is bigger: 10-3 or 10-2?[2 marks]
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10-2
10-3 = 1 ÷ 1000 = 0.001. 10-2 = 1 ÷ 100 = 0.01. 0.01 is bigger. Answer is 10-2.
C · Fractional (Higher)
- 11Evaluate 491/2[2 marks]
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7
The square root of 49 is the number that, when multiplied by itself, gives 49. That number is 7. - 12Evaluate 641/3[2 marks]
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4
The cube root of 64 is the number that, when multiplied by itself three times, gives 64. That number is 4. - 13Evaluate 163/4[2 marks]
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8
First find the fourth root of 16, which is 2. Then cube it: 2 × 2 × 2 = 8. - 14Evaluate 8-2/3[2 marks]
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1/4
First find the cube root of 8, which is 2. Then take the reciprocal and square it: 1/2 × 1/2 = 1/4. - 15Evaluate (9/25)3/2[2 marks]
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27/125
First find the square root of 9/25, which is 3/5. Then cube it: 3/5 × 3/5 × 3/5 = 27/125.
D · Algebra (Higher)
- 16Simplify (3a2b)3[2 marks]
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27a6b3
Multiply the power by each part: 33 = 27, a23 = a6, b3 = b3. Combine: 27a6b3 - 17Simplify (x-2)-3[2 marks]
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x6
Multiply the powers: -2 × -3 = 6. So x-2-3 = x6 - 18Simplify 6x5 ÷ 2x2[2 marks]
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3x3
Divide coefficients: 6 ÷ 2 = 3. Subtract powers: x5-2 = x3. So 3x3 - 19Simplify (16x8)1/2[2 marks]
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4x4
Take square root of 16 = 4 and x81/2 = x4. So 4x4
E · Solve and show (Higher)
- 20Find n when 2^n = 32.[3 marks]
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5
2^n = 32. 25 = 32. So n = 5. - 21Solve 3^x = 1/9.[3 marks]
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-2
3^x = 1/9. 1/9 = 3-2. So x = -2. - 22Write 85 as a power of 2.[3 marks]
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215
8 = 23. So 85 = (23)5 = 23×5 = 215. - 23Show that 43 × 2-2 = 16.[3 marks]
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16
43 = (22)3 = 26. 26 × 2-2 = 26-2 = 24 = 16.
Indices (powers) exam checklist
- Check the base is the same before using the laws.
- Add powers when multiplying, subtract when dividing.
- Multiply the powers when raising a power to another power.
- Anything to the power of 0 is always 1.
- Negative powers mean you flip the fraction and make it positive.
- The denominator of a fractional power is the root you take first.
Got stuck on a question?
Find the part of this guide it belongs to and go through the worked example again, line by line. Then put your answer back into the question: if it doesn't work, look for a sign or a step you skipped.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
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