Grade 9 questions on exponents are presented along with solutions and detailed explanations .

## Rules and Properties of Exponents

The exponential form is a convenient way to write long repeated multiplications of the same number by itself. $\underbrace{ a \cdot a \cdot a ... a}_{ n \; \; \text{times} } = a^n$ $$a$$ is called the base and is a real number and $$n$$ is called the exponent and is an integer. $$a^n$$ is read " $$a$$ to the power $$n$$"

 Definitions and Names of Rules Rule Examples 1 Exponent form $$\underbrace{ a \cdot a \cdot a ... a}_{ n \; \; \text{times} } = a^n$$ $$4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4^5$$ $$2^3 = 2 \cdot 2 \cdot 2 = 8$$ 2 Negative Exponent $$a^{-n} = \dfrac{1}{a^n}$$ or   $$a^{n} = \dfrac{1}{a^{-n}}$$ $$3^{-4} = \dfrac{1}{3^4}$$ $$5^{-2} = \dfrac{1}{5^2}$$ 3 Product Rule with Same Base $$a^m \cdot a^n = a^{m+n}$$ $$2^4 \cdot 2^6 = 2^{4+6} = 2^{10}$$ $$3^{2 + 6 } = 3^2 \cdot 3^6$$ 4 Product Rule with Same Exponent $$a^m \cdot b^m = (a \cdot b)^m$$ $$2^5 \cdot 3^5 = (2 \cdot 3)^5 = 6^5$$ $$(4 \cdot 3)^2 = 4^2 \cdot 3^2$$ 5 Quotient Rule with Same Base $$\dfrac{a^m}{a^n} = a^{m - n}$$ $$\dfrac{2^6}{2^4} = 2^{6-4} = 2^{2}$$ $$3^{5 - 2 } = \dfrac{3^5}{3^2}$$ 6 Quotient Rule with Same Exponent $$\left( \dfrac{a}{b} \right)^m = \dfrac{a^m}{b^m}$$ $$\left( \dfrac{3}{5} \right)^4 = \dfrac{3^4}{5^4}$$ $$\dfrac{4^2}{5^2} = \left( \dfrac{4}{5} \right)^2$$ 7 Quotient Rule with Negative Exponent $$\left( \dfrac{a}{b} \right)^{-m} = \dfrac{b^m}{a^m}$$ $$\left( \dfrac{3}{5} \right)^{-2} = \dfrac{5^2}{3^2}$$ 8 Power Rule $$(a^n)^m = a^{n \cdot m}$$ $$(2^3)^4 = 2^{3\cdot4} = 2^{12}$$ $$3^{4 \cdot 5} = (3^4)^5 = (3^5)^4$$ 9 Exponent zero Rule $$a^0 = 1 , \text{for} a \ne 0$$ $$10000000^0 = 1$$ $$1 = 2^0 = 8^0 = 12090^0$$ NOTE $$\color{red}{{ 0^0 = \text{undefined}}}$$ 10 Exponent one Rule $$a^1 = a$$ $$45^1 = 45$$ $$100 = 100^1$$ $$7 = 7^1$$ 11 Base one Rule $$1^n = 1$$ $$1^{230} = 1$$ $$1^{-100} = 1$$ 12 Negative one in Base Rule $$(-1)^n =\begin{cases} 1 , \text{if n even} \\ -1 , \text{if n odd} \end{cases}$$ $$(-1)^{19} = -1$$ $$(-1)^{18} = 1$$

## Questions

DO NOT USE A CALCULATOR.

1. Evaluate the following.

1. ) $$1^1$$
2. ) $$2^3$$
3. ) $$(-2)^2$$
4. ) $$(-2)^3$$
5. ) $$3^4$$
6. ) $$4^2$$
7. ) $$2^5$$
8. ) $$5^2$$
9. ) $$(-1)^6$$
10. ) $$7^2$$
11. ) $$(-9)^2$$
12. ) $$3^3$$
13. ) $$10^2$$
14. ) $$10^3$$
15. ) $$0.1^3$$

2. Write the following numbers in exponential form with exponent not equal to $$1$$. There might be more than one answer.

1. ) $$0$$
2. ) $$1$$
3. ) $$4$$
4. ) $$8$$
5. ) $$9$$
6. ) $$16$$
7. ) $$25$$
8. ) $$32$$
9. ) $$49$$
10. ) $$64$$
11. ) $$81$$
12. ) $$100$$
13. ) $$-27$$
14. ) $$-8$$
15. ) $$-64$$

3. Use the above rules to evaluate the following expressions.

1. ) $$120^0$$
2. ) $$2^{-3}$$
3. ) $$2^{-3} \cdot 2^6$$
4. ) $$2^{3} \cdot 3^3$$
5. ) $$\dfrac{3^{10}}{3^8}$$
6. ) $$4^{-1}$$
7. ) $$\dfrac{8^3}{4^3}$$
8. ) $$\dfrac{100^3}{10^3}$$
9. ) $$(2^{2})^2$$
10. ) $$(1^{3})^{25}$$
11. ) $$((-1)^{2})^{20}$$
12. ) $$- 2^{-2}$$
13. ) $$((-1)^{-1})^{-1}$$
14. ) $$\left (\dfrac{100}{10} \right)^{-2}$$
15. ) $$\left (\dfrac{10}{1000} \right)^{-2}$$

4. Simplify and write the following expressions with a single positive exponent if possible.

1. ) $$3^2 \cdot 3^8$$
2. ) $$\dfrac{2^5}{2^2}$$
3. ) $$\left( {3^5} \right)^2$$
4. ) $$6^4 \cdot \dfrac{6^5}{6^2}$$
5. ) $$(-7)^2 \cdot (-7)^3$$
6. ) $$\left( {5^2} \right)^2 \cdot \left( {5^3} \right)^3 \cdot 5$$
7. ) $$x^{-1} x^3$$
8. ) $$\dfrac{a^5}{a^2}$$
9. ) $$\dfrac{a^2}{a^7}$$
10. ) $$2^{x} \cdot 4^3 \cdot 2^y$$
11. ) $$(3^{-1})^x$$
12. ) $$3^{x} \cdot 9^{x}$$
13. ) $$\dfrac{a^x}{a^4} a^6$$

5. Simplify following expressions.

1. ) $$a^2 \cdot \dfrac{a^5}{a^2}$$
2. ) $$\left (\dfrac{3x}{x} \right)^3$$
3. ) $$(2^{2})^2$$
4. ) $$\dfrac{1}{4} \cdot \left (\dfrac{2x}{x} \right)^2$$
5. ) $$\dfrac{y^4 x^3}{x^2y^2}$$
6. ) $$\dfrac{x^2}{4y^2} \cdot \left (\dfrac{8y}{x} \right)^2$$
7. ) $$6 a^2 \cdot (a^2 + 1)^0$$

Solutions and detailed explanations to the above questions.