5.2: The Product Rule for Exponents
For any real number \(a\) and positive numbers \(m\) and \(n\), the product rule for exponents is the following.
\(a^m \cdot a^n = a^{m+n}\)
Note: Bases must be the same to use the product rule.
Idea:
From the last section, \(x^3 = \textcolor{blue}{ x \cdot x \cdot x }\qquad x^5 = \textcolor{red}{x \cdot x \cdot x \cdot x \cdot x}\)
Their product
\(x^3 \cdot x^5 = \textcolor{blue}{x \cdot x \cdot x} \textcolor{red}{\cdot x \cdot x \cdot x \cdot x \cdot x} = x^8\)
Hence, \(x^3 \cdot x^5 = x^{3+5 }= x^8\)
Use the product rule of exponents to simplify expressions.
- \(k^3 \cdot k^9\)
- \(\left(\dfrac{2 }{7}\right)^2 \cdot \left(\dfrac{2 }{7}\right)^6\)
- \((−2a)^3 \cdot (−2a)^7\)
- \(x \cdot x^3 \cdot x^{11}\)
- \(y^{13 }\cdot y^{33}\)
- \(x^3 \cdot y^2 \cdot x \cdot y^4\)
Solution
| Expression | Product Rule | Base |
| \(k^3 \cdot k^9\) | \(k^{3+9}= k^{12}\) | \(k\) |
| \(\left(\dfrac{2 }{7}\right)^2 \cdot \left(\dfrac{2 }{7}\right)^6\) | \(\left( \dfrac{2 }{7}\right)^{2+6 }= \left(\dfrac{2 }{7}\right)^8\) | \(\dfrac{2}{7}\) |
| \((−2a)^3 \cdot (−2a)^7\) | \((−2a)^{3+7 }= (−2a)^{10}\) | \(-2a\) |
| \(x \cdot x^3 \cdot x^{11}\) | \(x ^{1+3+11 }= x^{15}\) | \(x\) |
| \(y^{13 }\cdot y^{33}\) | \(y^{13+33 }= y^46\) | \(y\) |
| \(x^3 \cdot y^2 \cdot x \cdot y^4\) | \(x^{3+1 }\cdot y ^{2+4 }= x^{ 4 }\cdot y^{6}\) | \(x\) and \(y\) |
Note: Again, the bases MUST be the same to simplify using the product rule of exponent
Helpful steps to simplify using the product rule of exponents:
- Identify terms with common bases
- Identify the exponent of common bases.
- Add exponents of common bases and make the result of the sum the new exponent.
- Repeat steps as need
Use the product rule of exponents to simplify the following.
- \(f^3 \cdot f^11\)
- \(\left(\dfrac{x}{7}\right)^2 \cdot \left(\dfrac{x }{7}\right)^3\)
- \((−7x)^9 \cdot (−7x)^7\)
- \(h^5 \cdot h^3 \cdot h^{11}\)
- \(t^{13} \cdot t^{33}\)
- \(x^8 \cdot y^2 \cdot z \cdot x^ 3 \cdot y^2 \cdot z^{17}\)
- \(x^3 \cdot y^4 \cdot x^3\)