Indices worksheet - differentiated - Free Printable

To solve the given problems involving the laws of indices, we will use the following key rules:

1. Product Rule: \( a^m \times a^n = a^{m+n} \)
2. Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
3. Power Rule: \( (a^m)^n = a^{m \cdot n} \)
4. Zero Exponent Rule: \( a^0 = 1 \) (for \( a \neq 0 \))
5. Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)

Let's solve each problem step by step.

---

Section A: Simplify the following

#### A) \( 3^2 \times 3^5 \)
Using the Product Rule:
\[
3^2 \times 3^5 = 3^{2+5} = 3^7
\]

#### B) \( 4^3 \times 4^2 \)
Using the Product Rule:
\[
4^3 \times 4^2 = 4^{3+2} = 4^5
\]

#### C) \( y^8 \times y^3 \)
Using the Product Rule:
\[
y^8 \times y^3 = y^{8+3} = y^{11}
\]

#### D) \( (-2)^3 \times (-2)^2 \)
Using the Product Rule:
\[
(-2)^3 \times (-2)^2 = (-2)^{3+2} = (-2)^5
\]
Since \((-2)^5\) is negative:
\[
(-2)^5 = -32
\]

#### E) \( 3^3 \times 3^4 \)
Using the Product Rule:
\[
3^3 \times 3^4 = 3^{3+4} = 3^7
\]

#### F) \( (a^5)^3 \)
Using the Power Rule:
\[
(a^5)^3 = a^{5 \cdot 3} = a^{15}
\]

#### G) \( (z^2)^3 \)
Using the Power Rule:
\[
(z^2)^3 = z^{2 \cdot 3} = z^6
\]

#### H) \( (2c^3)^3 \)
Using the Power Rule for both the coefficient and the variable:
\[
(2c^3)^3 = 2^3 \cdot (c^3)^3 = 8 \cdot c^{3 \cdot 3} = 8c^9
\]

#### I) \( (ab^2)^2 \)
Using the Power Rule for both \( a \) and \( b^2 \):
\[
(ab^2)^2 = a^2 \cdot (b^2)^2 = a^2 \cdot b^{2 \cdot 2} = a^2b^4
\]

#### J) \( (2b^3)^2 \times (2b^4)^4 \)
First, apply the Power Rule to each term:
\[
(2b^3)^2 = 2^2 \cdot (b^3)^2 = 4b^6
\]
\[
(2b^4)^4 = 2^4 \cdot (b^4)^4 = 16b^{16}
\]
Now, multiply the results:
\[
4b^6 \times 16b^{16} = (4 \cdot 16) \cdot (b^6 \cdot b^{16}) = 64b^{6+16} = 64b^{22}
\]

#### K) \( (5^3)^2 \times (5^4)^3 \)
First, apply the Power Rule to each term:
\[
(5^3)^2 = 5^{3 \cdot 2} = 5^6
\]
\[
(5^4)^3 = 5^{4 \cdot 3} = 5^{12}
\]
Now, multiply the results:
\[
5^6 \times 5^{12} = 5^{6+12} = 5^{18}
\]

#### L) \( z^5 \div z^2 \)
Using the Quotient Rule:
\[
z^5 \div z^2 = z^{5-2} = z^3
\]

#### M) \( 8^1 \div 8^3 \)
Using the Quotient Rule:
\[
8^1 \div 8^3 = 8^{1-3} = 8^{-2}
\]
Using the Negative Exponent Rule:
\[
8^{-2} = \frac{1}{8^2} = \frac{1}{64}
\]

#### N) \( z^3 \div z^4 \)
Using the Quotient Rule:
\[
z^3 \div z^4 = z^{3-4} = z^{-1}
\]
Using the Negative Exponent Rule:
\[
z^{-1} = \frac{1}{z}
\]

#### O) \( 6a^3 \div 2a \)
Simplify the coefficients and apply the Quotient Rule for the variables:
\[
6a^3 \div 2a = \frac{6}{2} \cdot \frac{a^3}{a} = 3 \cdot a^{3-1} = 3a^2
\]

#### P) \( 15m^3k \div 3m^2k^4 \)
Simplify the coefficients and apply the Quotient Rule for the variables:
\[
15m^3k \div 3m^2k^4 = \frac{15}{3} \cdot \frac{m^3}{m^2} \cdot \frac{k}{k^4} = 5 \cdot m^{3-2} \cdot k^{1-4} = 5m^1k^{-3}
\]
Using the Negative Exponent Rule:
\[
5m^1k^{-3} = \frac{5m}{k^3}
\]

#### Q) \( (ab^3)^2 \div ab^2 \)
First, expand \( (ab^3)^2 \):
\[
(ab^3)^2 = a^2 \cdot (b^3)^2 = a^2b^6
\]
Now, divide:
\[
\frac{a^2b^6}{ab^2} = \frac{a^2}{a} \cdot \frac{b^6}{b^2} = a^{2-1} \cdot b^{6-2} = a^1b^4 = ab^4
\]

#### R) \( c^2 \times c^6 \div c \)
First, simplify the multiplication using the Product Rule:
\[
c^2 \times c^6 = c^{2+6} = c^8
\]
Now, divide:
\[
c^8 \div c = c^{8-1} = c^7
\]

#### S) \( 27c^2 \times c^1 \div 9c^4 \)
First, simplify the multiplication using the Product Rule:
\[
27c^2 \times c^1 = 27c^{2+1} = 27c^3
\]
Now, divide:
\[
27c^3 \div 9c^4 = \frac{27}{9} \cdot \frac{c^3}{c^4} = 3 \cdot c^{3-4} = 3c^{-1}
\]
Using the Negative Exponent Rule:
\[
3c^{-1} = \frac{3}{c}
\]

#### T) \( 3ab^2 \times 6a^2b^2 \)
Multiply the coefficients and apply the Product Rule for the variables:
\[
3ab^2 \times 6a^2b^2 = (3 \cdot 6) \cdot (a \cdot a^2) \cdot (b^2 \cdot b^2) = 18a^{1+2}b^{2+2} = 18a^3b^4
\]

#### U) \( -2x^3y^4 \times -2x^2y^4 \)
Multiply the coefficients and apply the Product Rule for the variables:
\[
-2x^3y^4 \times -2x^2y^4 = (-2 \cdot -2) \cdot (x^3 \cdot x^2) \cdot (y^4 \cdot y^4) = 4x^{3+2}y^{4+4} = 4x^5y^8
\]

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Final Answers

\[
\boxed{
\begin{array}{ll}
\text{A)} & 3^7 \\
\text{B)} & 4^5 \\
\text{C)} & y^{11} \\
\text{D)} & -32 \\
\text{E)} & 3^7 \\
\text{F)} & a^{15} \\
\text{G)} & z^6 \\
\text{H)} & 8c^9 \\
\text{I)} & a^2b^4 \\
\text{J)} & 64b^{22} \\
\text{K)} & 5^{18} \\
\text{L)} & z^3 \\
\text{M)} & \frac{1}{64} \\
\text{N)} & \frac{1}{z} \\
\text{O)} & 3a^2 \\
\text{P)} & \frac{5m}{k^3} \\
\text{Q)} & ab^4 \\
\text{R)} & c^7 \\
\text{S)} & \frac{3}{c} \\
\text{T)} & 18a^3b^4 \\
\text{U)} & 4x^5y^8 \\
\end{array}
}
\]