Using Fractional Exponents and Re-writing as Roots and Powers
Description/Explanation/Highlights
Video Description
In this video we learn the use and meaning of fractional exponents and also how to re-write fractional exponents into roots and powers form.
Steps and Key Points to Remember
To use fractional exponents, follow these steps:
- When an exponent appears as a fraction, the top (numerator) of the fraction represents the exponent while the bottom (denominator) of the fraction represents the root. i.e. \( 16^{\frac{1}{2}}=\sqrt[2]{16^1}=\sqrt{16}\)
- When finding an answer, order doesn’t matter. You can find the root first and then raise that answer to a power or you can raise to the power first and then find the root of that answer. To find \(16^{\frac{3}{4}}\) take \(\sqrt[4]{16} = 2 \) then \(2^3=8\) OR take \(16^3=4,096 \) then \(\sqrt[4]{4,096}=8 \)
- When a fractional exponent is negative, move the base and the power to the denominator before changing to a root and power i.e. \(16^{\frac{-3}{4}}=\frac{1}{16^{\frac{3}{4}}}=\frac{1}{8} \)
- When simplifying problems with fractional exponents, apply rules of exponents such as adding exponents when multiplying like bases or subtracting exponents when dividing like bases just like you would with integer exponents. i.e. \(\frac{48x^{\frac{13}{2}}y^{\frac{5}{2}}}{12x^{\frac{9}{2}}y^{\frac{3}{2}}}=4x^2y \) by dividing the lead coefficients and subtracting and simplifying the exponents of x and y.
- When problems with fractional exponents cannot be solved without a calculator, the problem is often rewritten to roots and powers and left in that form i.e. \(7^{\frac{5}{8}}=\sqrt[8]{7^5} \)
Here are some key points to keep in mind when working with fractional exponents:
- The numerator of a fractional exponent represents the power and the denominator represents the root.
- When simplifying and finding answers, order does not matter. You can apply the power first and then take the root or take the root first and then apply the power. The answer will be the same.
- Always simplify complex problems by applying the rules of exponents before changing to roots and powers.
- Rules of simplifying fractional exponents are the same as they are for integer exponents.
- It is often easier to solve a problem after changing to roots and powers if the root is applied first and then the power. This results in dealing with smaller numbers. This only works if the root is easy to to find unless a calculator is used.
Video Highlights
- 00:00 Introduction to using fractional exponents
- 00:17 \(16^{\frac{1}{2}}=\sqrt[2]{16^1}\) example of changing fractional exponents to roots and powers
- 01:42 \(16^{\frac{3}{4}}=\sqrt[4]{16^3}=8\) example of changing fractional exponents to roots and powers and solving
- 02:50 \(16^{\frac{-3}{4}}=\frac{1}{8} \) example using a negative fractional exponent
- 03:45 \( \frac{48x^{\frac{13}{2}}y^{\frac{5}{2}}}{12x^{\frac{9}{2}}y^{\frac{3}{2}}}=4x^2y \) example of simplifying an exponential problem with fractional exponents
- 05:10 \(7^{\frac{5}{8}}=\sqrt[8]{7^5} \)example of when a problem cannot be solved without a calculator after changing to roots and powers