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Exponents Questions with Answers for Grade 9

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 and is a real number and $n$ is called the 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	$a^0 = 1 , \text{for} a \ne 0$	$10000000^0 = 1$ $1 = 2^0 = 8^0 = 12090^0$ $\color{red}{{ 0^0 = \text{undefined}}}$
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.

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. ) $0.1^3$
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. ) $-64$
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. ) $\left (\dfrac{10}{1000} \right)^{-2}$
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. ) $\dfrac{a^x}{a^4} a^6$
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.

Exponents Questions with Answers for Grade 9

Rules and Properties of Exponents

Questions

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