Learning objective
We have previously learned about the product property for exponents. Let's review where it comes from.
Consider the expression a^{5} \cdot a^{3}. Let's think abou what it would look like if we expanded it:
We can see how this could be rewritten as a^8. Notice, 8 is also the sum of the powers in the original expression.
So, in our example above, \begin{aligned}a^{5}\cdot a^{3} &= a^{5+3}\\&=a^{8}\end{aligned}
This leads us to the product property for exponents:
b^{m} \cdot b^{n}=b^{m+n}
That is, when multiplying terms with a common base:
Keep the same base
Find the sum of the exponents
In other words, when multiplying terms with like bases, we add the exponents.
We can visualize this graphically as a transformation. Consider a simple exponential function: f\left(x\right)=2^x.
If we apply a translation to the left 3 units we can write the function as f\left(x\right)=2^{\left(x+3\right)}.
Applying the product property we can rewrite the function as f\left(x+3\right)=2^{\left(x+3\right)}=2^x\cdot 2^3=8\cdot2^x
Fill in the blank to make the eqution true: b^{2}\cdot b^{⬚} = b^{2 + 3}
Simplify m^{2} \cdot m^{7} + r^{3} \cdot r^{2}, giving your answer in exponential form.
Consider the function: h\left(x\right)=2^{\left(x-4\right)}
Identify the transformation on the parent function that produced h\left(x\right).
Rewrite h\left(x\right) to use a vertical dilation instead of a translation.
We can use the product property of exponents: b^{m} \cdot b^{n}=b^{m+n} to rewrite a horizontal translation as a vertical dilation and vice versa.
We have previously learned about the power property for exponents. Let's review where it comes from.
Consider the expression (a^{2})^{3}. What is the resulting power of base a? To find out, have a look at the expanded form of the expression:
\displaystyle (a^{2})^{3} | \displaystyle = | \displaystyle (a^{2})\times (a^{2}) \times (a^{2}) |
\displaystyle = | \displaystyle (a \times a) \times (a \times a) \times (a \times a) | |
\displaystyle = | \displaystyle a \times a \times a \times a \times a \times a | |
\displaystyle = | \displaystyle a^{6} |
In the expanded form, we can see that we are multiplying six groups of a together. That is, \\(a^{2})^{3}=a^{6}.
We can confirm this result using the product property for exponents:
\displaystyle (a^{2}) \times (a^{2}) \times (a^{2}) | \displaystyle = | \displaystyle a^{2+2+2} |
\displaystyle = | \displaystyle a^6 |
This leads us to the power property for exponents:
(b^{m})^{n} = b^{m\cdot n}
That is, when simplifying a term with a power that itself has a power:
Keep the same base
Multiply the exponents
We can visualize this graphically as a transformation. Consider again the simple exponential function: f\left(x\right)=2^x.
If we apply a horizontal dilation by a factor of 3 we can write the function as f\left(3x\right)=2^{\left(3x\right)}.
Applying the power property we can rewrite the function as f\left(3x\right)=2^{\left(3x\right)}={\left(2^3\right)}^x=8^x
Simplify \left(a^{5}\right)^{3}.
Simplify (a^{9}\cdot b^{3})^{4}
Simplify \left(-2x^{2}\right)^{2}.
We can use the power property of exponents: \left(b^m\right)^m=b^{m\cdot n} to rewrite a horizontal dilation as an exponential function with a different base.
The negative exponent property states: b^{-n}=\dfrac{1}{b^{n}}
This can be understood as a 'flipping' operation, where the base moves from the numerator to the denominator, or vice versa, when the sign of the exponent changes. In other words, if you have a negative exponent, you can make it positive by taking the reciprocal of the base.
Similarly, an exponential expression can involve what is called an exponential unit fraction. This is when the exponent is the reciprocal of a natural number (positive whole number), denoted as \dfrac{1}{k}. An exponential unit fraction takes the form b^{\frac{1}{k}}.
Let's consider the example x^{\frac{1}{2}}:
\displaystyle x^{\frac{2}{2}} | \displaystyle = | \displaystyle x |
\displaystyle \left(x^{\frac{1}{2}}\right)^{2} | \displaystyle = | \displaystyle x |
\displaystyle \left(x^{\frac{1}{2}}\right)^{2} | \displaystyle = | \displaystyle \left(\sqrt{x}\right)^{2} |
\displaystyle x^{\frac{1}{2}} | \displaystyle = | \displaystyle \sqrt{x} |
An exponential unit fraction, b^{\frac{1}{k}}, represents the kth root of the base b.
Express 3x^{-3} with a positive exponent.
Consider the following.
Rewrite x^{\frac{1}{3}} in radical form.
Evaluate \sqrt[3]{x} for when x=8.
Write the following with a fractional index: \sqrt[7]{72}
Negative exponent property: b^{-n}=\dfrac{1}{b^{n}}
Exponential unit fractions:
x^{\frac{1}{n}}=\sqrt[n]{x}