5.02 Power rule
Power rule
Exploration
Move the sliders m and n to change the number of exponents. Click on the checkboxes to see working steps and answers.
Move the slider to change the base's exponent (m) to 2 and the exponent outside the parentheses (n) to 4.
After clicking "Show expanded form", how many groups of a^2's are there?
Click "Show answer". What do you notice about the exponent in the simplified form?
Adjust the sliders to change the base's exponent and the exponent outside the parentheses. What happens to the simplified form \left(a^m\right)^n?
The power rule states that for any base number, a, and any numbers m and n as power, \left(a^{m}\right)^{n} = a^{m\cdot n}
That is, when simplifying a term with a power that itself has a power:
Keep the same base
Multiply the exponents
\left(a^2\right)^{3} = \left(a^2\right) \left(a^2\right) \left(a^2\right)
\left(a^2\right)^{3} = \left(a \cdot a\right)\left(a \cdot a\right)\left(a \cdot a\right)
\left(a^2\right)^{3} = a \cdot a \cdot a \cdot a \cdot a \cdot a
\left(a^2\right)^{3} = a^6
When using the power rule for exponents, the coefficient is handled separately from the exponents. Let's take a look at an example.
\left(3x^2\right)^4
Raise the coefficient to the power: 3^4 =81
Multiply the exponents, applying the power rule: \left(x^2\right)^4 = x^{2\cdot 4} = x^8
Therefore, \left(3x^2\right)^4 simplifies to: 81x^8
Examples
Example 1
Simplify \left(a^{5}\right)^{3}.
For any base number a, and any numbers m and n as power, \left(a^{m}\right)^{n} = a^{m\cdot n}
That is, when simplifying a term with a power that itself has a power:
Keep the same base
Multiply the exponents
Power of a product rule
Exploration
Consider the following expressions:
\begin{aligned} \left(2 \cdot 3 \right)^{2} \quad \text{and} \quad 2^{2} \cdot 3^{2} \\ \left(4 \cdot 5 \right)^{3} \quad \text{and} \quad 4^{3} \cdot 5^{3} \\ \left(7 \cdot 2 \right)^{4} \quad \text{and} \quad 7^{4} \cdot 2^{4} \\ \left(3 \cdot 6 \right)^{5} \quad \text{and} \quad 3^{5} \cdot 6^{5} \end{aligned}
Evaluate each expression and record your results in the table:
| Expression | Result |
|---|---|
| (2 \cdot 3)^2 | |
| 2^2 \cdot 3^2 | |
| (4 \cdot 5)^3 | |
| 4^3 \cdot 5^3 | |
| (7 \cdot 2)^4 | |
| 7^4 \cdot 2^4 | |
| (3 \cdot 6)^5 | |
| 3^5 \cdot 6^5 |
- What patterns do you notice in the results of the expressions?
-
Can you form a rule or law based on your observations?
For the product of any numbers a and b in the base, and for any number n in the power, \left(ab\right)^{n}=a^{n}b^{n}
The power of a product rule states that a product raised to a power is equivalent to the product of the two factors each raised to the same power.
(ab)^{4} = (a^4) (b^4)
(ab)^{4} = (a \cdot a \cdot a \cdot a)(b \cdot b \cdot b \cdot b)
(ab)^{4} = a^4 b^4
Examples
Example 2
Simplify \left(a^{9}\cdot b^{3}\right)^{4}
Example 3
Simplify \left(-2x^{2}\right)^{2}.
The power of a product rule states that for the product of any numbers a and b in the base, and for any number n in the power, (ab)^{n}=a^{n}b^{n}
In other words, a product raised to a power is equivalent to the product of the two factors each raised to the same power.