Create an Index Worksheet using Excel Hyperlinks - Excel University - Free Printable

Problem: Simplify the given expressions using the rules of indices.

#### Rules of Indices Recap:
1. \( a^m \times a^n = a^{m+n} \)
2. \( a^m \div a^n = a^{m-n} \)
3. \( (a^m)^n = a^{mn} \)
4. \( a^{-n} = \frac{1}{a^n} \)
5. \( a^{\frac{1}{n}} = \sqrt[n]{a} \)
6. \( a^0 = 1 \)
7. \( a^1 = a \)

---

Section 1: Simplify the following expressions

#### Part (a): \( x^4 \times x^3 \)
Using the rule \( a^m \times a^n = a^{m+n} \):
\[
x^4 \times x^3 = x^{4+3} = x^7
\]

#### Part (b): \( y^5 \times y^2 \)
Using the rule \( a^m \times a^n = a^{m+n} \):
\[
y^5 \times y^2 = y^{5+2} = y^7
\]

#### Part (c): \( p^3 \times p \)
Using the rule \( a^m \times a^n = a^{m+n} \):
\[
p^3 \times p = p^3 \times p^1 = p^{3+1} = p^4
\]

#### Part (d): \( x^7 \times x^{-3} \)
Using the rule \( a^m \times a^n = a^{m+n} \):
\[
x^7 \times x^{-3} = x^{7 + (-3)} = x^{7-3} = x^4
\]

#### Part (e): \( x^5 \div x^2 \)
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
x^5 \div x^2 = x^{5-2} = x^3
\]

#### Part (f): \( \frac{y^7}{y^3} \)
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
\frac{y^7}{y^3} = y^{7-3} = y^4
\]

#### Part (g): \( n^{10} \div n^4 \)
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
n^{10} \div n^4 = n^{10-4} = n^6
\]

#### Part (h): \( z^6 \div z^3 \)
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
z^6 \div z^3 = z^{6-3} = z^3
\]

#### Part (i): \( x^5 \div x^{-2} \)
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
x^5 \div x^{-2} = x^{5 - (-2)} = x^{5+2} = x^7
\]

#### Part (j): \( \frac{w^5}{w^{-3}} \)
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
\frac{w^5}{w^{-3}} = w^{5 - (-3)} = w^{5+3} = w^8
\]

#### Part (k): \( p^5 \div p^{-2} \)
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
p^5 \div p^{-2} = p^{5 - (-2)} = p^{5+2} = p^7
\]

#### Part (l): \( x^3 \div x^{-3} \)
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
x^3 \div x^{-3} = x^{3 - (-3)} = x^{3+3} = x^6
\]

#### Part (m): \( (x^4)^2 \)
Using the rule \( (a^m)^n = a^{mn} \):
\[
(x^4)^2 = x^{4 \cdot 2} = x^8
\]

#### Part (n): \( (y^7)^{-2} \)
Using the rule \( (a^m)^n = a^{mn} \):
\[
(y^7)^{-2} = y^{7 \cdot (-2)} = y^{-14}
\]
Using the rule \( a^{-n} = \frac{1}{a^n} \):
\[
y^{-14} = \frac{1}{y^{14}}
\]

#### Part (o): \( (z^{-1})^{-3} \)
Using the rule \( (a^m)^n = a^{mn} \):
\[
(z^{-1})^{-3} = z^{(-1) \cdot (-3)} = z^3
\]

#### Part (p): \( \left( \frac{1}{w^2} \right)^{-3} \)
Using the rule \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \):
\[
\left( \frac{1}{w^2} \right)^{-3} = \frac{1^{-3}}{(w^2)^{-3}}
\]
Since \( 1^{-3} = 1 \):
\[
\frac{1^{-3}}{(w^2)^{-3}} = \frac{1}{(w^2)^{-3}}
\]
Using the rule \( (a^m)^n = a^{mn} \):
\[
(w^2)^{-3} = w^{2 \cdot (-3)} = w^{-6}
\]
Using the rule \( a^{-n} = \frac{1}{a^n} \):
\[
\frac{1}{w^{-6}} = w^6
\]

---

Section 2: Simplify the following expressions

#### Part (a): \( (ab)^4 \)
Using the rule \( (ab)^n = a^n b^n \):
\[
(ab)^4 = a^4 b^4
\]

#### Part (b): \( (xy)^3 \)
Using the rule \( (ab)^n = a^n b^n \):
\[
(xy)^3 = x^3 y^3
\]

#### Part (c): \( \frac{3}{(3a)^2} \)
First, simplify \( (3a)^2 \):
\[
(3a)^2 = 3^2 \cdot a^2 = 9a^2
\]
So:
\[
\frac{3}{(3a)^2} = \frac{3}{9a^2} = \frac{1}{3a^2}
\]

#### Part (d): \( (2a)^3 \)
Using the rule \( (ab)^n = a^n b^n \):
\[
(2a)^3 = 2^3 \cdot a^3 = 8a^3
\]

#### Part (e): \( (3y)^4 \)
Using the rule \( (ab)^n = a^n b^n \):
\[
(3y)^4 = 3^4 \cdot y^4 = 81y^4
\]

#### Part (f): \( (mn^2)^5 \)
Using the rule \( (ab)^n = a^n b^n \):
\[
(mn^2)^5 = m^5 \cdot (n^2)^5
\]
Using the rule \( (a^m)^n = a^{mn} \):
\[
(n^2)^5 = n^{2 \cdot 5} = n^{10}
\]
So:
\[
(mn^2)^5 = m^5 n^{10}
\]

---

Section 3: Simplify the following expressions

#### Part (a): \( 2a^2 \times a^3 \)
First, simplify \( a^2 \times a^3 \):
\[
a^2 \times a^3 = a^{2+3} = a^5
\]
So:
\[
2a^2 \times a^3 = 2a^5
\]

#### Part (b): \( x^2 \times 3x^4 \)
First, simplify \( x^2 \times x^4 \):
\[
x^2 \times x^4 = x^{2+4} = x^6
\]
So:
\[
x^2 \times 3x^4 = 3x^6
\]

#### Part (c): \( 2y^2 \times 3y^2 \)
First, simplify \( y^2 \times y^2 \):
\[
y^2 \times y^2 = y^{2+2} = y^4
\]
So:
\[
2y^2 \times 3y^2 = (2 \cdot 3)y^4 = 6y^4
\]

#### Part (d): \( 8x^2 \div 2x \)
First, simplify \( 8 \div 2 \):
\[
8 \div 2 = 4
\]
Next, simplify \( x^2 \div x \):
\[
x^2 \div x = x^{2-1} = x^1 = x
\]
So:
\[
8x^2 \div 2x = 4x
\]

#### Part (e): \( 10y^3 \div 2y^2 \)
First, simplify \( 10 \div 2 \):
\[
10 \div 2 = 5
\]
Next, simplify \( y^3 \div y^2 \):
\[
y^3 \div y^2 = y^{3-2} = y^1 = y
\]
So:
\[
10y^3 \div 2y^2 = 5y
\]

#### Part (f): \( 6z^4 \div 3z^2 \)
First, simplify \( 6 \div 3 \):
\[
6 \div 3 = 2
\]
Next, simplify \( z^4 \div z^2 \):
\[
z^4 \div z^2 = z^{4-2} = z^2
\]
So:
\[
6z^4 \div 3z^2 = 2z^2
\]

#### Part (g): \( 10y^2 \div 5y^3 \)
First, simplify \( 10 \div 5 \):
\[
10 \div 5 = 2
\]
Next, simplify \( y^2 \div y^3 \):
\[
y^2 \div y^3 = y^{2-3} = y^{-1} = \frac{1}{y}
\]
So:
\[
10y^2 \div 5y^3 = 2 \cdot \frac{1}{y} = \frac{2}{y}
\]

#### Part (h): \( 12x^4 \div 3x^{-2} \)
First, simplify \( 12 \div 3 \):
\[
12 \div 3 = 4
\]
Next, simplify \( x^4 \div x^{-2} \):
\[
x^4 \div x^{-2} = x^{4 - (-2)} = x^{4+2} = x^6
\]
So:
\[
12x^4 \div 3x^{-2} = 4x^6
\]

---

Section 4: Multiply out the brackets

#### Part (a): \( x^3 \left( x^2 + x^4 \right) \)
Distribute \( x^3 \):
\[
x^3 \left( x^2 + x^4 \right) = x^3 \cdot x^2 + x^3 \cdot x^4
\]
Using the rule \( a^m \times a^n = a^{m+n} \):
\[
x^3 \cdot x^2 = x^{3+2} = x^5
\]
\[
x^3 \cdot x^4 = x^{3+4} = x^7
\]
So:
\[
x^3 \left( x^2 + x^4 \right) = x^5 + x^7
\]

#### Part (b): \( y^2 \left( y^3 - y \right) \)
Distribute \( y^2 \):
\[
y^2 \left( y^3 - y \right) = y^2 \cdot y^3 - y^2 \cdot y
\]
Using the rule \( a^m \times a^n = a^{m+n} \):
\[
y^2 \cdot y^3 = y^{2+3} = y^5
\]
\[
y^2 \cdot y = y^{2+1} = y^3
\]
So:
\[
y^2 \left( y^3 - y \right) = y^5 - y^3
\]

#### Part (c): \( z^2 \left( z^4 - 1 \right) \)
Distribute \( z^2 \):
\[
z^2 \left( z^4 - 1 \right) = z^2 \cdot z^4 - z^2 \cdot 1
\]
Using the rule \( a^m \times a^n = a^{m+n} \):
\[
z^2 \cdot z^4 = z^{2+4} = z^6
\]
\[
z^2 \cdot 1 = z^2
\]
So:
\[
z^2 \left( z^4 - 1 \right) = z^6 - z^2
\]

---

Section 5: Simplify

#### Part (a): \( \frac{x^3 \times x^4}{x^2} \)
First, simplify the numerator \( x^3 \times x^4 \):
\[
x^3 \times x^4 = x^{3+4} = x^7
\]
So:
\[
\frac{x^3 \times x^4}{x^2} = \frac{x^7}{x^2}
\]
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
\frac{x^7}{x^2} = x^{7-2} = x^5
\]

#### Part (b): \( \frac{t^3 \times t^6}{t^4} \)
First, simplify the numerator \( t^3 \times t^6 \):
\[
t^3 \times t^6 = t^{3+6} = t^9
\]
So:
\[
\frac{t^3 \times t^6}{t^4} = \frac{t^9}{t^4}
\]
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
\frac{t^9}{t^4} = t^{9-4} = t^5
\]

#### Part (c): \( \frac{w^2 \times w^{-3}}{w^{-4}} \)
First, simplify the numerator \( w^2 \times w^{-3} \):
\[
w^2 \times w^{-3} = w^{2 + (-3)} = w^{2-3} = w^{-1}
\]
So:
\[
\frac{w^2 \times w^{-3}}{w^{-4}} = \frac{w^{-1}}{w^{-4}}
\]
Using the rule \( a^m \div a^n = a^{m-n} \):
\[
\frac{w^{-1}}{w^{-4}} = w^{-1 - (-4)} = w^{-1+4} = w^3
\]

---

Final Answers:

1.
- (a) \( x^7 \)
- (b) \( y^7 \)
- (c) \( p^4 \)
- (d) \( x^4 \)
- (e) \( x^3 \)
- (f) \( y^4 \)
- (g) \( n^6 \)
- (h) \( z^3 \)
- (i) \( x^7 \)
- (j) \( w^8 \)
- (k) \( p^7 \)
- (l) \( x^6 \)
- (m) \( x^8 \)
- (n) \( \frac{1}{y^{14}} \)
- (o) \( z^3 \)
- (p) \( w^6 \)

2.
- (a) \( a^4 b^4 \)
- (b) \( x^3 y^3 \)
- (c) \( \frac{1}{3a^2} \)
- (d) \( 8a^3 \)
- (e) \( 81y^4 \)
- (f) \( m^5 n^{10} \)

3.
- (a) \( 2a^5 \)
- (b) \( 3x^6 \)
- (c) \( 6y^4 \)
- (d) \( 4x \)
- (e) \( 5y \)
- (f) \( 2z^2 \)
- (g) \( \frac{2}{y} \)
- (h) \( 4x^6 \)

4.
- (a) \( x^5 + x^7 \)
- (b) \( y^5 - y^3 \)
- (c) \( z^6 - z^2 \)

5.
- (a) \( x^5 \)
- (b) \( t^5 \)
- (c) \( w^3 \)

\boxed{
\begin{aligned}
&\text{1. (a) } x^7, \text{ (b) } y^7, \text{ (c) } p^4, \text{ (d) } x^4, \text{ (e) } x^3, \text{ (f) } y^4, \text{ (g) } n^6, \text{ (h) } z^3, \text{ (i) } x^7, \text{ (j) } w^8, \text{ (k) } p^7, \text{ (l) } x^6, \text{ (m) } x^8, \text{ (n) } \frac{1}{y^{14}}, \text{ (o) } z^3, \text{ (p) } w^6 \\
&\text{2. (a) } a^4 b^4, \text{ (b) } x^3 y^3, \text{ (c) } \frac{1}{3a^2}, \text{ (d) } 8a^3, \text{ (e) } 81y^4, \text{ (f) } m^5 n^{10} \\
&\text{3. (a) } 2a^5, \text{ (b) } 3x^6, \text{ (c) } 6y^4, \text{ (d) } 4x, \text{ (e) } 5y, \text{ (f) } 2z^2, \text{ (g) } \frac{2}{y}, \text{ (h) } 4x^6 \\
&\text{4. (a) } x^5 + x^7, \text{ (b) } y^5 - y^3, \text{ (c) } z^6 - z^2 \\
&\text{5. (a) } x^5, \text{ (b) } t^5, \text{ (c) } w^3
\end{aligned}
}