Exponents in Math Explained | Grade 7 Examples and Practice

Exponents are a fundamental concept in mathematics that show how many times a number, called the base, is multiplied by itself. On this page, you will find Grade 7 math examples and algebra questions involving exponents, along with clear, step-by-step solutions.

These practice problems cover topics such as writing expressions with powers, evaluating exponents, and simplifying using exponent rules.

What are exponents in math and where are they used?

Exponents are a way to represent repeated multiplication of a number by itself. The formula is written as:

\[ \Large{a \times a \times a \times \cdots \times a \quad (n \text{ times}) = a^n} \]

Real-World Examples of Exponents

Example 1: Basic Exponentiation

\[ 5 \times 5 \times 5 = 5^3 \]

Here:

  • Base = \(5\)
  • Exponent = \(3\)
  • Meaning: Multiply 5 by itself 3 times.

Example 2: Representing Large Numbers

\[ 100{,}000 = 10 \times 10 \times 10 \times 10 \times 10 = 10^5 \]

Example 3: Representing Small Numbers

\[ 0.00001 = \frac{1}{100{,}000} = \frac{1}{10^5} = 10^{-5} \]

A negative exponent means taking the reciprocal. For example:

\[ 10^{-2} = \frac{1}{10^2} = \frac{1}{100} \]

Example 4 & 5: Units of Area and Volume

Area: The area of a square with a side of \(1\) meter is:
\[ 1 \,\text{m} \times 1 \,\text{m} = 1 \,\text{m}^2 \] (Read as: 1 square meter)

Volume: The volume of a cube with a side of \(1\) meter is:
\[ 1 \,\text{m} \times 1 \,\text{m} \times 1 \,\text{m} = 1 \,\text{m}^3 \] (Read as: 1 cubic meter)

Example 6: Scientific Prefixes

  • Kilo (\(10^3\)): \( 1{,}000 = 10^3 \)
    (e.g., 1 kilogram = \(10^3\) grams = \(1{,}000\) grams)
  • Mega (\(10^6\)): \( 1{,}000{,}000 = 10^6 \)
    (e.g., 1 megabyte = \(10^6\) bytes = \(1{,}000{,}000\) bytes)
  • Giga (\(10^9\)): \( 1{,}000{,}000{,}000 = 10^9 \)
    (e.g., 1 gigabyte = \(10^9\) bytes = \(1{,}000{,}000{,}000\) bytes)

Exponent Practice Problems & Solutions

Solve the following exponents problems. Expand the solution blocks to check your reasoning and the final answers.

Question 1

Write the following using exponents:

  1. \( 8 \times 8 \times 8 \times 8 \)
  2. \( 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \)
  3. \( A \times A \times A \)
  4. \( \text{meter} \times \text{meter} \)
  5. \( \text{centimeter} \times \text{centimeter} \times \text{centimeter} \)
View Solution

Use the definition of an exponent, where the base is the repeated factor and the exponent is the number of times it is multiplied by itself.

  1. \( 8 \times 8 \times 8 \times 8 = \mathbf{8^4} \)
    (8 multiplied by itself 4 times)
  2. \( 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = \mathbf{10^7} \)
  3. \( A \times A \times A = \mathbf{A^3} \)
  4. \( \text{meter} \times \text{meter} = \mathbf{\text{meter}^2} \)
  5. \( \text{centimeter} \times \text{centimeter} \times \text{centimeter} = \mathbf{\text{centimeter}^3} \)

Question 2

Evaluate the following:

  1. \( 2^4 \)
  2. \( 10^4 \)
  3. \( (-2)^4 \)
  4. \( -2^4 \)
View Solution

Expand each power using multiplication and calculate the result. Pay special attention to the location of the negative signs in (c) and (d).

  1. \( 2^4 = 2 \times 2 \times 2 \times 2 = \mathbf{16} \)
  2. \( 10^4 = 10 \times 10 \times 10 \times 10 = \mathbf{10{,}000} \)
  3. \( (-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = \mathbf{16} \)
    (The negative sign is included in the base because of the parentheses. Multiplying four negative numbers results in a positive.)
  4. \( -2^4 = -(2 \times 2 \times 2 \times 2) = \mathbf{-16} \)
    (There are no parentheses, so the exponent only applies to the 2. The negative sign is applied after exponentiation.)

Question 3

Use exponents to write the following using one power only:

  1. \( 4 \times 8 \)
  2. \( 25 \times 5 \)
  3. \( 16 \times 4 \times 4^3 \)
  4. \( 2 \times 2 \times 8 \times 2^3 \)
  5. \( B \times B \times B^3 \)
View Solution

To combine into one power, break down the numbers into a common base, then add the exponents.

  1. \( 4 \times 8 = (2 \times 2) \times (2 \times 2 \times 2) = 2^2 \times 2^3 = \mathbf{2^5} \)
  2. \( 25 \times 5 = (5 \times 5) \times 5 = 5^2 \times 5^1 = \mathbf{5^3} \)
  3. \( 16 \times 4 \times 4^3 = (4 \times 4) \times 4^1 \times 4^3 = 4^2 \times 4^1 \times 4^3 = \mathbf{4^6} \)
  4. \( 2 \times 2 \times 8 \times 2^3 = 2^1 \times 2^1 \times (2 \times 2 \times 2) \times 2^3 = 2^1 \times 2^1 \times 2^3 \times 2^3 = \mathbf{2^8} \)
  5. \( B \times B \times B^3 = B^1 \times B^1 \times B^3 = \mathbf{B^5} \)

Question 4

Use exponents to rewrite the following expressions in simplified form:

  1. \( 2^3 \times 2^4 \)
  2. \( 6 \times 6^3 \)
  3. \( 5 \times 5^2 \times 5^3 \)
View Solution

When multiplying powers with the same base, you simply add the exponents (Product Rule: \( a^m \times a^n = a^{m+n} \)).

  1. \( 2^3 \times 2^4 = 2^{3+4} = \mathbf{2^7} \)
    (Expanded proof: \((2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^7\))
  2. \( 6 \times 6^3 = 6^1 \times 6^3 = 6^{1+3} = \mathbf{6^4} \)
  3. \( 5 \times 5^2 \times 5^3 = 5^1 \times 5^2 \times 5^3 = 5^{1+2+3} = \mathbf{5^6} \)

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