Laws of integral indices · Ages 13–14
Powers of powers, products and quotients
(am)n multiplies the indices; every factor in the bracket is raised.
How to do it
Method
(a^m)^n multiplies the indices; every factor in the bracket is raised
(x3)4 = x12; (2x3)2 = 4x6
Another way
Write it as a product
(x3)4 = x3 × x3 × x3 × x3 = x12
In full
A power of a power: multiply the indices, (am)n = a^(mn). (x3)4 is four x3's multiplied: x3 × x3 × x3 × x3 = x^(3 + 3 + 3 + 3) = x12, so 3 × 4 = 12. A power of a product: every factor is raised, (ab)n = an bn. So (2x3)2 = 22 × (x3)2 = 4x6: the 2 is squared too, it is not 2x6. A power of a quotient: the top and the bottom are both raised, (ab)n = anbn, so (x3)2 = x29. This is for multiplying and dividing only: (a + b)2 is not a2 + b2. Or write the power out as a product and multiply powers of the same base: the answer is the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Fill in the index: (x8)4 = x□
Show the answer and working
Answer: 32
4 lots of x8 multiplied: multiply the indices, 8 × 4 = 32.
Example 2
Simplify: (5x7)3 = x□
Fill in each blank
Show the answer and working
Answer: 125, 21
Every factor in the bracket is raised: 53 = 125 and (x7)3 = x21.
Example 3
(2x2)3 = ?
- A. 2x6
- B. 8x5
- C. 6x6
- D. 8x6
Choose the right answer
Show the answer and working
Answer: D. 8x6
Raise every factor: 23 = 8 and (x2)3 = x6, so 8x6.
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: (x5)5 = x□
Answer:
2.Fill in the index: (108)3 = 10□
Answer:
3.(3x2)3 = ?
- A. 3x6
- B. 27x5
- C. 9x6
- D. 27x6
Choose the right answer
Answer:
4.Simplify: (x2y6)2 = x□y□
Fill in each blank
Answer:
5.(2x6)3 = 2x18
True or false?
Answer:
All the answers
- 1. 25
- 2. 24
- 3. D. 27x6
- 4. 4, 12
- 5. False
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
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