Laws of integral indices · Ages 13–14
Writing powers with the same base
One base first, then the laws; same base, equal indices.
How to do it
Method
One base first, then the laws; same base, equal indices
43 × 82 = 26 × 26 = 212
Another way
Work it out to check
43 × 82 = 64 × 64 = 4096 = 212
In full
When the bases differ, see whether they can be written as powers of one base: 4 = 22, 8 = 23, 16 = 24, 9 = 32, 27 = 33, 25 = 52, 18 = 2−3. Take 43 × 82: 43 = (22)3 = 26 and 82 = (23)2 = 26, so 43 × 82 = 26 × 26 = 212. Equations with the unknown in the index go the same way: 2x = 32, and 32 = 25. With the same base, the indices are equal, so x = 5. For 9x = 27, 3^(2x) = 33, so 2x = 3 and x = 32. The slip to watch: multiplying the bases, 43 × 82 is not 325. When the numbers are small, work them out to check: 43 × 82 = 64 × 64 = 4096 = 212.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
As a power of 5: 1256 = 5□
Show the answer and working
Answer: 18
125 = 53, so 1256 = (53)6 = 518.
Example 2
274 × 92 = (3□)4 × (3□)2 = 3□
Fill in each blank
Show the answer and working
Answer: 3, 2, 16
27 = 33 and 9 = 32; the index: 3 × 4 + 2 × 2 = 16.
Example 3
94 × 813 = 320
True or false?
Show the answer and working
Answer: True
One base first: 94 × 813 = 3^(2 × 4 + 4 × 3) = 320; the bases can't be multiplied with the indices added.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.As a power of 5: 253 = 5□
Answer:
2.Solve: 81x = 3, x =
Answer in simplest form
Answer:
3.275 × 814 = ?
- A. 21879
- B. 39
- C. 279
- D. 331
Choose the right answer
Answer:
4.As a power of 2: 23 × 163 = 2□
Answer:
5.274 × 92 = 316
True or false?
Answer:
All the answers
- 1. 6
- 2. 14
- 3. D. 331
- 4. 15
- 5. True
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