Laws of integral indices · Ages 13–14
Multiplying and dividing powers
Same base: multiplying adds the indices, dividing subtracts them.
How to do it
Method
Same base: multiplying adds the indices, dividing subtracts them
x4 × x5 = x9; x7 ÷ x3 = x4
Another way
Write them all out
x3 × x2 = (x × x × x) × (x × x) = x5
In full
Powers of the same base multiplied: add the indices, am × an = a^(m + n). The reason is plain: a3 × a2 = (a × a × a) × (a × a), five a's multiplied, which is a5. Powers of the same base divided: subtract the indices, am ÷ an = a^(m − n) (m bigger than n). a5 ÷ a2 is five a's multiplied, divided by two a's; cancel the two and three are left, a3. With numbers in front, multiply or divide those as usual, and add or subtract the indices of each letter: 3x4 × 2x5 = 6x9. Only the same base combines like this: 23 × 32 is not 65, and x3 + x3 is 2x3, not x6. Or write every power out as a product and count the equal factors: the answer is the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Simplify: 10x6 ÷ 2x4 = x□
Fill in each blank
Show the answer and working
Answer: 5, 2
Divide the numbers: 10 ÷ 2 = 5; subtract the indices: 6 − 4 = 2. So 5x2.
Example 2
Simplify: 2x2 × 9x3 = x□
Fill in each blank
Show the answer and working
Answer: 18, 5
Multiply the numbers: 2 × 9 = 18; add the indices: 2 + 3 = 5. So 18x5.
Example 3
x7 × x6 ÷ x3 = ?
- A. x16
- B. x39
- C. x13
- D. x10
Choose the right answer
Show the answer and working
Answer: D. x10
Left to right: the index is 7 + 6 − 3 = 10, so x10.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Fill in the index: x2 × x2 = x□
Answer:
2.Fill in the index: 72 × 73 = 7□
Answer:
3.Simplify: 3x7 × 3x8 = x□
Fill in each blank
Answer:
4.Fill in the index: 38 × 36 ÷ 310 = 3□
Answer:
5.Simplify: 5x4y4 × 3x7y3 = x□y□
Fill in each blank
Answer:
All the answers
- 1. 4
- 2. 5
- 3. 9, 15
- 4. 4
- 5. 15, 11, 7
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake