Grade 8 · Exponents
Working with Exponents
Meaning of exponents · product rule · quotient rule · zero and negative exponents
Section 1 of 5
Read repeated multiplication, identify the base and exponent, and expand expressions
Key Rule
An exponent tells you how many times the base is used as a factor.
\(a^n\) means multiply \(a\) by itself \(n\) times.
Base and Exponent
Example
1
In \(5^3\), the base is 5 and the exponent is 3.
2
This means \(5 \times 5 \times 5\).
Common mistake: \(2^4\) does not mean \(2 \times 4\). The exponent tells you how many times to multiply the base by itself.
Check Your Understanding
What does \(3^4\) mean?
In \(7^2\), what is the base?
Calculate: \(4^3\). Type your answer.
\(4^3 = 4 \times 4 \times 4 = 64\)
Multiplying Powers with the Same Base
Use the product rule: when bases match, add the exponents
Key Rule
\(a^m \times a^n = a^{m+n}\)
Keep the base the same. Add the exponents.
Example · \(2^3 \times 2^5\)
1
The bases are the same: both are 2.
2
Add the exponents: \(3 + 5 = 8\).
3
So \(2^3 \times 2^5 = 2^8\) = \(256\)
Common mistake: You can only add exponents when the bases are the same. \(2^3 \times 3^3\) does not become \(5^6\).
Check Your Understanding
Simplify: \(x^2 \times x^5\)
Simplify: \(6^1 \times 6^3\)
Calculate: \(2^4 \times 2^2\). Type your answer.
\(2^{4+2} = 2^6 = 64\)
Dividing Powers with the Same Base
Use the quotient rule: when bases match, subtract the exponents
Key Rule
\(a^m \div a^n = a^{m-n}\) \((a \neq 0)\)
Keep the base the same. Subtract the exponents.
Example · \(8^6 \div 8^2\)
2
Subtract exponents: \(6 - 2 = 4\)
Common mistake: Do not subtract the bases. Only the exponents change when the bases are the same.
Check Your Understanding
Simplify: \(a^8 \div a^3\)
Simplify: \(10^7 \div 10^2\)
Calculate: \(x^6 \div x^6\). Type your answer.
\(x^{6-6} = x^0\), and \(x^0 = 1\) for \(x \neq 0\)
Power of a Power, Zero Exponents, and Negative Exponents
Learn the last core exponent rules used in Grade 8
Power of a Power
\((a^m)^n = a^{m \times n}\)
Multiply the exponents.
Example · \((3^2)^4\)
1
Multiply the exponents: \(2 \times 4 = 8\)
3
If needed, the value is \(6561\)
Zero Exponent Rule
\(a^0 = 1\) \((a \neq 0)\)
Any non-zero number raised to the power 0 equals 1.
Example · \(9^0\)
1
Because the base is not zero, \(9^0 = 1\).
Negative Exponent Rule
\(a^{-n} = \frac{1}{a^n}\) \((a \neq 0)\)
A negative exponent means take the reciprocal.
Example · \(2^{-3}\)
1
Rewrite as a fraction: \(\frac{1}{2^3}\)
2
Calculate: \(\frac{1}{8}\)
3
Final answer: \(\frac{1}{8}\)
Common mistake: \(a^{-n}\) is not negative. It creates a reciprocal. For example, \(5^{-2} = \frac{1}{25}\).
Check Your Understanding
Simplify: \((x^2)^3\)
What is \(5^0\)?
Calculate: \(3^{-2}\). Type your answer.
\(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)
Practice Quiz
10 questions covering exponent meaning and the main laws of exponents