This document provides examples and explanations of rules for manipulating expressions with indices or exponents. It includes the basic rules like adding or subtracting exponents with the same base, multiplying expressions with the same base, and dividing/multiplying expressions with the same base. The document then provides practice problems applying these index laws to simplify expressions and solve equations.

1. D. J. Jayamanne
NSBM 1
Indices

2. Recap: Rules of Indices
Indices are a useful way of more simply expressing large numbers.
To manipulate expressions, we can consider using the Law of Indices.
These laws only apply to expressions with the same base
Rule1 𝑎𝑚
× 𝑎𝑛
= 𝑎(𝑚+𝑛)
Rule2 (𝑎𝑚
)𝑛
= 𝑎𝑚×𝑛
Rule3 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎(𝑚−𝑛)
Rule4 𝑎𝑚 =
1
𝑎−𝑚 and 𝑎−𝑚=
1
𝑎𝑚
Rule5 𝑎0
= 1
Rule6 𝑎
1
𝑚 = 𝑚
𝑎
NSBM 2

3. Answer the following questions
Q1) Find the simplified form of the following. Each expression should have
positive exponents.
i. (𝑝5)4
ii. (𝑝4
)5
iii. (𝑝−5)4
iv. 𝑦3(𝑦−5)2
v. (4𝑚)3
NSBM 3

4. Questions
Q2)
2.1 Find the area of a square whose side is 5𝑥3
2.2 Write an expression for the perimeter of the above square
2.3 Find the volume of a cube whose side is 2𝑥2
Q3) Simplify the following. Express each expression in positive exponents.
i. 𝑛5 2(4𝑚𝑛−2)
ii. (𝑥−2)2 3𝑥𝑦5 4
iii. 3𝑐5 4
𝑐2 3
iv. 6𝑎𝑏 3 5𝑎−3 2
NSBM 4

5. Questions
Q4) Simplify the following:
4.1 4.2
NSBM 5
i. 𝑛3 6
ii. 𝑏−7 3
iii. 3𝑎 4
iv. 9𝑥5 2 𝑥2 5
v. 4 × 105 2
vi. 2 × 10−3 5
vii. 𝑛8 4
viii. 𝑛4 8
ix. 𝑐2 5
x. 𝑞10 10
i. 𝑤7 −1
ii. 𝑥3 −5
iii. 𝑑 𝑑−2 −9
iv. 𝑧8 0
𝑧5
v. 𝑎5 3𝑐4
vi. 𝑐3 5 𝑑3 0
vii. 𝑡2 −2
𝑡2 −5
viii. 𝑚3 −1
𝑥2 5

6. Questions
Q5) Simplify each expression
5.1 5.2
NSBM 6
i. 4𝑚 5
ii. 7𝑎 −2
iii. 5𝑦 4
iv. 12𝑔4 −1
v. 3𝑛−6 −4
vi. 2𝑦4 −3
vii. 𝑥𝑦 0
viii. 𝑟2
𝑠 5
ix. 2𝑥 3𝑥2
x. 𝑦2
𝑧−3 5
𝑦3 2
i. 𝑚𝑔4 −1 𝑚𝑔4
ii. 𝑝 𝑝−7
𝑞3 −2
𝑞−3
iii. 3𝑏−2 2
𝑎2
𝑏4 3
iv. 𝑐−12 𝑐−2𝑑 3𝑑5
v. 2𝑗2𝑘4 −5 𝑘−1𝑗7 6
vi. 4𝑗2
𝑘6
(2𝑗11
)3
𝑘5

7. Questions
Q6) Simplify. Write each answer in scientific notation.
Note: Scientific notation is a compact way of writing small / large
numbers. NSBM 7
i. 3 × 105 2
ii. 4 × 102 5
iii. 2 × 10−10 3
iv. 2 × 10−3 3
v. 7.4 × 104 2
vi. 6.25 × 10−12 −2
vii. 3.5 × 10−4 3
viii. 2.37 × 108 3

8. Questions
Q7) Complete each equation.
NSBM 8
i. 𝑏2  = 𝑏8
ii. 𝑚
3
= 𝑚−12
iii. 𝑥
7
= 𝑥6
iv. 𝑛9  = 1
v. 𝑦−4  = 𝑦12
vi. 7 𝑐1  = 7𝑐8
vii. 5𝑥
2
= 25𝑥−4
viii. 3𝑥3𝑦
3
= 27𝑥9
ix. 𝑚2𝑛3  =
1
𝑚6𝑛9

9. Questions
Q8) Simplify each expression.
NSBM 9
i. 32 3𝑥 3
ii. (4.1)5 4.1 −5
iii. (𝑏5)3𝑏2
iv. (−5𝑥)2
+5𝑥2
v. (−2𝑎2𝑏)3(𝑎𝑏)3
vi. (2𝑥−3)2(0.2𝑥)2
vii. 4x𝑦204(−𝑦)−3
viii. 103 4
(4.3 × 10−8
)
ix. (37)2(3−4)3

10. Questions
Q9) Can you write 49𝑥2𝑦2𝑧2 using one exponent?
Show how or why not?
Q10) Can you write
27
64
𝑥6𝑦−3𝑧3 using one exponent?
Show how or why not?
NSBM 10

11. Questions
Q11) Solve each equation. Use the fact that if 𝑎𝑥=𝑎𝑦 then
𝑥 = 𝑦.
NSBM 11
i. 5𝑥 = 25𝑥
ii. 3𝑥 = 274
iii. 82 = 2𝑥
iv. 4𝑥 = 26
v. 32𝑥
= 94
vi. 2𝑥 =
1
32

12. Questions
Q12) What is the simplified form of each expression?
NSBM 12
i.
𝑦5
𝑦4
ii.
𝑑3
𝑑9
iii.
𝑘6𝑗2
𝑘𝑗5
iv.
𝑎−3𝑏7
𝑎5𝑏2
v.
𝑥4𝑦−1𝑧8
𝑧𝑥4𝑦5

13. Questions
Q13) Show that
𝑎
𝑏
−𝑛
=
𝑏
𝑎
𝑛
for nonzero numbers 𝑎 and 𝑏 and positive
integers 𝑛.
Q14) Simplify each expression:
NSBM 13
i.
𝑎
5𝑏
−2
ii.
2𝑥6
𝑦4
−3
iii.
𝑚
𝑛
−3
iv.
3𝑥2
5𝑦4
−4

14. Questions
Q15) Simplify each expression:
NSBM 14
i.
38
36
ii.
36
38
iii.
𝑑14
𝑑17
iv.
𝑛−1
𝑛−4
v.
5𝑎−7
10𝑎−9
vi.
𝑥11𝑦3
𝑥11𝑦
vii.
𝑐3𝑑−5
𝑐4𝑑−1
viii.
10𝑚6𝑛3
5𝑚2𝑛7
ix.
𝑚3𝑛2
𝑚−1𝑛3
x.
𝑎−3
𝑎−9

15. Questions
Q16) Simplify each expression:
NSBM 15
i.
32𝑚5𝑡6
35𝑚7𝑡−5
ii.
𝑥5𝑦−8𝑧3
𝑥𝑦−4𝑧3
iii.
12𝑎−1𝑏6𝑐−3
4𝑎5𝑏−1𝑐5