Area of a Circle — Summary
Let's look more closely at the final area of the circle and the final volume of the sphere to see if we can find a more efficient way than the product rule to simplify these expressions.
Area of a Circle
\[\begin{align*} A&=\pi(x^3)^2\\ &=\pi x^6 \end{align*}\]
Volume of a Sphere
\[\begin{align*} V&=\frac{4}{3}\pi(x^7)^3\\ &=\frac{4}{3}\pi x^{21} \end{align*}\]
In each equation, we have a power raised to an exponent. This is referred to as a power of a power.
Comparing the area of a circle and volume of a sphere formulas side by side, can you come up with the rule for simplifying a power of a power?
You may notice that if we multiply the exponents together on each power of a power, we can quickly simplify the expression.
- If we revisit \((x^3)^2\) and apply the multiplication, we end up with \(x^{(3)(2)}\) or \(x^6\).
- For the power of a power \((x^7)^3\) within the sphere's volume calculation, we can show this simplification as \(x^{(7)(3)}\) or \(x^{21}\).
The power of a power rule states that when a power is raised to an exponent, you may multiply the exponents together to simplify the expression.
\[(x^a)^b=x^{(a)(b)}\]
Try This Revisited
At the start of this lesson, we were looking at two expressions to decide whether they were equivalent or not. Let's take a closer look the following two expressions.
- \((4^2)(4^3)\)
- \((4^2)^3\)
Simplifying the two expressions, we can see
- \((4^2)(4^3)=4^{2+3}=4^5\), and
- \((4^2)^3=(4^2)(4^2)(4^2)=4^{2+2+2}=4^6\), or using the power of a power rule, \((4^2)^3 = 4^{(2)(3)}=4^6\).
Therefore, \((4^2)(4^3) \ne(4^2)^3\).
Although both expressions use similar numbers, they do not equal one another.
Consider the following two expressions:
- \((5^6)(5^6)\)
- \((5^6)^2\)
Are these expressions equal to one another?
Simplifying the two expressions, we can see
- \((5^6)(5^6)=5^{6+6}=5^{12}\), and
- \((5^6)^2=(5^6)(5^6)=5^{6+6}=5^{12}\), or using the power of a power rule, \((5^6)^2 = 5^{(6)(2)}=5^{12}\).
Here we can see that both expressions are equal, \((5^6)^2 = (5^6)(5^6)\).
Recall
When simplifying using the product rule for exponents, we add the exponents together, but when we are using the power of a power rule, we multiply the exponents by each other.