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 THE 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.