Lesson 5, Topic 5
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# Law 5: Zero Index

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This law states that any number raised to the power of zero (0) equals 1.

$$\scriptsize a^{0} = 1$$

$$\scriptsize m^{0} = 1$$

$$\scriptsize 10^{0} = 1$$

### Example

Simplify the following;

a. $$\scriptsize 4^{0}$$

b. $$\scriptsize\left( 5^6\right)^{0}$$

c. $$\scriptsize 5^2 \: \times \: 5^{-2}$$

d. $$\scriptsize 2^5 \: \div \: 2^{5}$$

e. $$\scriptsize 8a^0 \: \div \: 8^{0}$$

Solution

a. $$\scriptsize 4^{0} = 1$$

b. $$\scriptsize\left( 5^6\right)^{0} = 1$$

c. $$\scriptsize 5^2 \: \times \: 5^{-2} \\ \scriptsize =5^{2-2} \\ \scriptsize = 5^0 \\ \scriptsize = 1$$

d. $$\scriptsize 2^5 \: \div \: 2^{5} \\ \scriptsize = 2^{5-5} \\ \scriptsize = 2^0 \\ \scriptsize = 1$$

e. $$\scriptsize 8a^0 \: \div \: 8^{0} \\ \scriptsize 8 \: \times \: a^0 \: \div \: 8^o \\ \scriptsize = 8 \: \times \: 1 \: \div \: 1\\ \scriptsize = 8 \: \div \: 1 \\ \scriptsize = 8$$

Evaluation

1. Simplify the following by

(i) expansion of terms

(ii) law of indices

a. $$\scriptsize 3^4\: \times \: 3^2$$

b. $$\scriptsize a^3\: \times \: a^4$$

c. $$\scriptsize 5^3\: \times \: 5 \: \times \: 5^6$$

d. $$\left( \frac{3}{4} \right)^3 \: \times \: \left( \frac{3}{4} \right)^4$$

e. $$\scriptsize Z \: \times \: Z$$

f. $$\scriptsize x^5\: \times \: x \: \times \: x$$

g. $$\scriptsize c \: \times \: c\: \times \: c^2 \: \times \: c^3\: \times \: c^4$$

h. $$\scriptsize 8^2 \: \times \: 8^2 \: \times \: 8^3$$

2. Simplify the following

a. $$\scriptsize 2q^7 \: \times \: 3q^8 \: \times \: 4q$$

b. $$\normalsize \frac{2}{5} \scriptsize m^2 \: \times \: \normalsize \frac{5}{7} \scriptsize m^7$$

c. $$\scriptsize 8y^2 \: \times \: 7y^3$$

d. $$\scriptsize 7n^3 \: \times \: n^{\frac{1}{2}}$$

3. Simplify the following by:

i. expansion

ii. laws of indices

a. $$\scriptsize 15^5 \: \div \: 15^4$$

b. $$\scriptsize 4^7 \: \div \: 4^3$$

c. $$\scriptsize 3^{12} \: \div \: 3^7$$

d. $$\scriptsize 12^{5} \: \div \: 12$$

e. $$\scriptsize 20y^{14} \: \div \: 5y^5$$

f. $$\scriptsize 35d^{17} \: \div \: 7d^{10}$$

g. $$\scriptsize 3x^{3} \: \div \: 6x$$

h. $$\scriptsize 7d^{8} \: \div \: 38d^5$$

4. (A) Simplify the following by

i. expansion

ii. the law of negative index

a. $$\scriptsize 10^{-5}$$

b. $$\scriptsize 7^{-3}$$

c. $$\scriptsize 9^2 \: \div \: 9^4$$

d. $$\scriptsize x^5 \: \times \: x^{-7}$$

e. $$\scriptsize (1.5)^{-2}$$

f. $$\scriptsize 10^{12} \: \div \: 10^{15}$$

4(B) Simplify the following

a. $$\scriptsize 5a^{-3} \: \times \: 4b$$

b. $$\scriptsize (7x)^{-2}$$

c. $$\scriptsize 5a^{-1} \: \times \: a^{-3}$$

d. $$\scriptsize 2^{-3} \:\times \: 2^{-2} \: \times \: 2^4$$

e. $$\left(\frac{1}{8} \right)^{-2} \: \times \: \left( \frac{1}{2}\right)^3$$

f. $$\scriptsize 2g^6 \: \div \: 4g^7$$

5. Evaluate the following leaving your answer in index form when necessary.

a. $$\scriptsize (5^3)^5$$

b. $$\scriptsize ((2x)^4)^3$$

c. $$\scriptsize (0.5^2)^3$$

d. $$\left(\frac{1}{2y} \right)^{-2}$$

e. $$\left(\frac{5}{x} \right)^3$$

f. $$\scriptsize (5y)^3 \: \times \: y^{-10}$$

6. Simplify the following

a. $$\scriptsize 13^0$$

b. $$\scriptsize 1.3^0$$

c. $$\scriptsize 7^{-2} \: \times \: 7^2 \: \times \: 7^0$$

d. $$\scriptsize (2x)^0 \: \div \: x^3$$

e. $$\scriptsize 5x^2 \: \times \: (2x)^{-2}$$

f. $$\frac{(5^{10})^0}{5^4}$$

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