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 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.

More References and Links

Middle School Math (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
High School Math (Grades 10, 11 and 12) - Free Questions and Problems With Answers
Primary Math (Grades 4 and 5) with Free Questions and Problems With Answers Home Page
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