Simplify, with positive indices: x^2y^−7 × x^4y^−5 = x^□/y^□ Fill in each blank

6, 12. x's index: 2 + (4) = 6; y's: (−7) + (−5) = −12, and y^−12 = 1/y^12.

Simplify, with a positive index: (x^−3)^4 ÷ x^5 = ? A. 1/x^17 B. 1/x^12 C. x^17 D. x^7 Choose the right answer

A. 1/x^17. (−3) × 4 − 5 = −17, and x^−17 = 1/x^17.

With a positive index: (x^2/5)^−2 = □/x^□ Fill in each blank

25, 4. Turn the fraction over first: (x^2/5)^−2 = (5/x^2)^2 = 25/x^4.

Laws of integral indices · Ages 13–14

Working with integral indices

Work the indices as directed numbers, then make them positive.

How to do it

Method

Work the indices as directed numbers, then make them positive

x⁻³ × x⁵ = x²; (x⁻²)³ = x⁻⁶ = 1x⁶

Another way

Move it across the bar first

x² ÷ x⁻⁴ = x² × x⁴ = x⁶

In full

With zero and negative indices, every law of indices holds for any whole-number index: a^m × aⁿ = a^(m + n), a^m ÷ aⁿ = a^(m − n), (a^m)ⁿ = a^(mn), (ab)ⁿ = aⁿ bⁿ. Add, subtract and multiply the indices as directed numbers: x⁻³ × x⁵ = x^(−3 + 5) = x²; x² ÷ x⁻⁴ = x^(2 − (−4)) = x⁶; (x⁻²)³ = x⁻⁶. The answer is usually written with positive indices at the end: x⁻⁶ = 1x⁶, and 2x³y⁻² = 2x³y². Another way: move each factor with a negative index to the other side of the fraction bar first, where its index turns positive, then use the laws for positive indices. The answer is the same.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
Simplify, with positive indices: x²y⁻⁷ × x⁴y⁻⁵ = x^□y^□
Fill in each blank
Show the answer and working
Answer: 6, 12
x's index: 2 + (4) = 6; y's: (−7) + (−5) = −12, and y⁻¹² = 1y¹².
Example 2
Simplify, with a positive index: (x⁻³)⁴ ÷ x⁵ = ?
- A. 1x¹⁷
- B. 1x¹²
- C. x¹⁷
- D. x⁷
Choose the right answer
Show the answer and working
Answer: A. 1x¹⁷
(−3) × 4 − 5 = −17, and x⁻¹⁷ = 1x¹⁷.
Example 3
With a positive index: (x²5)⁻² = x^□
Fill in each blank
Show the answer and working
Answer: 25, 4
Turn the fraction over first: (x²5)⁻² = (5x²)² = 25x⁴.

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: x⁻⁸ × x⁻⁹ = x^□
Answer:
2.x³ ÷ x⁻⁹ = x⁻⁶
True or false?
Answer:
3.Fill in the index: (x³)⁻⁴ = x^□
Answer:
4.Fill in the indices: (x³y⁻³)⁻³ ÷ x⁵y⁻³ = x^□y^□
Fill in each blank
Answer:
5.(xy)⁻⁶ = x⁻⁶y⁻⁶
True or false?
Answer:

All the answers

1. −17
2. False
3. −12
4. −14, 12
5. True

Free worksheet: Working with integral indicesWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

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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