Topic Content:
- Law 5: Zero Index (Power of 0)
- Evaluation Questions
This law states that any number raised to the power of zero (0) equals 1.
Worked Example 5.5.1:
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. \( \scriptsize 8a^0 \: \div \: 8^{0} \\ \scriptsize 8 \: \times \: a^0 \: \div \: 8^0 \\ \scriptsize = 8 \: \times \: 1 \: \div \: 1\\ \scriptsize = 8 \: \div \: 1 \\ \scriptsize = 8\)
Evaluation Questions:
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. \( \scriptsize 8^2 \: \times \: 8^2 \: \times \: 8^3 \)
View Answers2. 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. \( \scriptsize 7n^3 \: \times \: n^{\frac{1}{2}}\)
View Answers3. 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. \(\scriptsize 7d^{8} \: \div \: 38d^5\)
View Answers4. (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. \( \scriptsize 10^{12} \: \div \: 10^{15} \)
View Answers4(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. \( \scriptsize 2g^6 \: \div \: 4g^7 \)
View Answers5. 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. \( \scriptsize (5y)^3 \: \times \: y^{-10} \)
View Answers6. 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. \( \frac{(5^{10})^0}{5^4} \)
View Answers