Laws of Indices interactive worksheet - Free Printable

Let’s solve each problem step by step.

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1. Evaluate:

a) \(2^6\)
This means multiply 2 by itself 6 times:
\(2 × 2 = 4\)
\(4 × 2 = 8\)
\(8 × 2 = 16\)
\(16 × 2 = 32\)
\(32 × 2 = 64\)
→ Answer: 64

b) \(219^0\)
Any number (except 0) raised to the power of 0 is 1.
→ Answer: 1

c) \(\left(\frac{1}{6}\right)^2\)
Square both numerator and denominator:
\(\frac{1^2}{6^2} = \frac{1}{36}\)
→ Answer: \(\frac{1}{36}\)

d) \(3^{-4}\)
Negative exponent means take reciprocal and make exponent positive:
\(3^{-4} = \frac{1}{3^4} = \frac{1}{81}\)
→ Answer: \(\frac{1}{81}\)

e) \(-5^3\)
Be careful! This means “negative of \(5^3\)”, not \((-5)^3\).
First, \(5^3 = 5 × 5 × 5 = 125\), then add the negative sign → -125
→ Answer: -125

f) \(\left(\frac{5}{3}\right)^{-2}\)
Negative exponent → flip the fraction and make exponent positive:
\(\left(\frac{3}{5}\right)^2 = \frac{9}{25}\)
→ Answer: \(\frac{9}{25}\)

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2. Simplify

We use rules:
- Multiply coefficients (numbers) together.
- For same base variables, add exponents when multiplying, subtract when dividing.

a) \(7w^2 × 4w^3\)
Multiply numbers: \(7 × 4 = 28\)
Add exponents for w: \(w^{2+3} = w^5\)
→ Answer: \(28w^5\)

b) \((3x^2y^3)(5xy^2)\)
Numbers: \(3 × 5 = 15\)
x: \(x^2 × x = x^{2+1} = x^3\)
y: \(y^3 × y^2 = y^{3+2} = y^5\)
→ Answer: \(15x^3y^5\)

c) \(\frac{2}{3}h^3 × \frac{5}{9}h\)
Numbers: \(\frac{2}{3} × \frac{5}{9} = \frac{10}{27}\)
h: \(h^3 × h = h^{3+1} = h^4\)
→ Answer: \(\frac{10}{27}h^4\)

d) \(3m^2 × 4m × 2\)
Numbers: \(3 × 4 × 2 = 24\)
m: \(m^2 × m = m^{2+1} = m^3\)
→ Answer: \(24m^3\)

e) \(3p^2 × p^4 × 2p^6\)
Numbers: \(3 × 1 × 2 = 6\) (note: \(p^4\) has coefficient 1)
p: \(p^2 × p^4 × p^6 = p^{2+4+6} = p^{12}\)
→ Answer: \(6p^{12}\)

f) \(2r^5 × 3r^2 × 4r^3\)
Numbers: \(2 × 3 × 4 = 24\)
r: \(r^5 × r^2 × r^3 = r^{5+2+3} = r^{10}\)
→ Answer: \(24r^{10}\)

g) \(8v^4 ÷ 2v\)
Numbers: \(8 ÷ 2 = 4\)
v: \(v^4 ÷ v = v^{4-1} = v^3\)
→ Answer: \(4v^3\)

h) \(r^5s^4 ÷ r^2s^3\)
r: \(r^{5-2} = r^3\)
s: \(s^{4-3} = s^1 = s\)
→ Answer: \(r^3s\)

i) \(21g^8h^2v ÷ 3gh\)
Numbers: \(21 ÷ 3 = 7\)
g: \(g^{8-1} = g^7\)
h: \(h^{2-1} = h^1 = h\)
v stays as is (no v in divisor)
→ Answer: \(7g^7hv\)

j) \(18y^{10}z^7 ÷ 9z^6\)
Numbers: \(18 ÷ 9 = 2\)
y: no change → \(y^{10}\)
z: \(z^{7-6} = z^1 = z\)
→ Answer: \(2y^{10}z\)

k) \(\frac{12g^{10}}{4g^7}\)
Numbers: \(12 ÷ 4 = 3\)
g: \(g^{10-7} = g^3\)
→ Answer: \(3g^3\)

l) \(\frac{6g^{11}h^9}{3g^4h^6}\)
Numbers: \(6 ÷ 3 = 2\)
g: \(g^{11-4} = g^7\)
h: \(h^{9-6} = h^3\)
→ Answer: \(2g^7h^3\)

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Final Answer:

1. Evaluate:
a) 64
b) 1
c) \(\frac{1}{36}\)
d) \(\frac{1}{81}\)
e) -125
f) \(\frac{9}{25}\)

2. Simplify:
a) \(28w^5\)
b) \(15x^3y^5\)
c) \(\frac{10}{27}h^4\)
d) \(24m^3\)
e) \(6p^{12}\)
f) \(24r^{10}\)
g) \(4v^3\)
h) \(r^3s\)
i) \(7g^7hv\)
j) \(2y^{10}z\)
k) \(3g^3\)
l) \(2g^7h^3\)