Directions: Using any number between 1 and 9, fill in the boxes to create a true statement. You may only use a number once.
Source: Open Middle |
In the Quadratics Unit, you learned how to simplify square roots, multiply and divide square roots, and add and subtract square roots. In this unit, you will be working with \(n^{th}\) roots, like a cube root, fourth root, etc.
The symbol we use for \(n^{th}\) roots is \(\sqrt[n]{a}\), where \(n\) is called the index and \(a\) is called the radicand. When you are simplifying an \(n^{th}\) root, you are trying to determine the principal root (the value the calculator would give). Basically, you are trying to find what value multiplied by itself \(n\) times will give you the radicand.
The symbol we use for \(n^{th}\) roots is \(\sqrt[n]{a}\), where \(n\) is called the index and \(a\) is called the radicand. When you are simplifying an \(n^{th}\) root, you are trying to determine the principal root (the value the calculator would give). Basically, you are trying to find what value multiplied by itself \(n\) times will give you the radicand.
Properties of \(n^{th}\) Roots for \(a>0\) and \(b>0\): Product Property: \(\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}\) Quotient Property: \(\large\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\) Power Property: \(\sqrt[n]{a^n}=\left(\sqrt[n]{a}\right)^{^n}=a\) |
Examples of Properties of \(n^{th}\) Roots for \(a>0\) and \(b>0\): Product Property: \(\sqrt[4]{2\cdot318}=\sqrt[4]{2}\cdot\sqrt[4]{318}\) Quotient Property: \(\large\sqrt[3]{\frac{81}{19}}=\frac{\sqrt[3]{81}}{\sqrt[3]{19}}\) Power Property: \(\sqrt[5]{4^5}=\left(\sqrt[5]{4}\right)^{^5}=4\) |
You can use these three properties (as illustrated in the above examples) to simplify \(n^{th}\) roots. An \(n^{th}\) root is fully simplified when the following three conditions are met:
- There are no perfect \(n^{th}\) power factors (other than \(1\)) in the radicand.
- There is no radical in the denominator of a fraction .
- The radicand is not a fraction.
Ideally you should be able to find the largest perfect \(n^{th}\) power factor of any radicand using mental math, but there are a few methods that will work if you are unable to find the correct factor quickly.
One method is to find the largest perfect \(n^{th}\) power factor of the radicand then apply the product property. A list of common values is shown below. Another method is to create a prime factor tree to write the radicand as a power of primes then apply the power property. Both methods are explained in the video examples, so you will probably want to watch the videos!
Table of common \(n^{th}\) powers
\(x\) |
\(x^2\) |
\(x^3\) |
\(x^4\) |
\(x^5\) |
\(x^6\) |
\(1\) |
\(1\) |
\(1\) |
\(1\) |
\(1\) |
\(1\) |
\(2\) |
\(4\) |
\(8\) |
\(16\) |
\(32\) |
\(64\) |
\(3\) |
\(9\) |
\(27\) |
\(81\) |
\(243\) |
\(729\) |
\(4\) |
\(16\) |
\(64\) |
\(256\) |
\(1024\) |
\(4096\) |
\(5\) |
\(25\) |
\(125\) |
\(625\) |
\(3125\) |
\(15625\) |
\(6\) |
\(36\) |
\(216\) |
\(1296\) |
\(7776\) |
\(46656\) |
\(7\) |
\(49\) |
\(343\) |
\(2401\) |
\(16807\) |
\(117649\) |
\(8\) |
\(64\) |
\(512\) |
\(4096\) |
\(32768\) |
\(262144\) |
\(9\) |
\(81\) |
\(729\) |
\(6561\) |
\(59049\) |
\(531441\) |
It is probably a good idea to recognize these values at least through the \(5^{th}\) power.
Example 1: Simplify \(\sqrt[3]{72}\).
Example 1: Simplify \(\sqrt[3]{72}\).
Example 2: Simplify \(\sqrt[4]{80}\).
Sometimes you need to apply the product property before you can simplify the \(n^{th}\) root (because one or both of the radicands do not have any perfect nth power factors) as illustrated below.
Example 3: Simplify \(\sqrt[5]{10}\cdot\sqrt[5]{64}\).
\(\begin{align}&\sqrt[5]{10}\cdot\sqrt[5]{64}\ & \ &\text{1) Rewrite original problem.}\\
&\sqrt[5]{10\cdot64}\ & \ &\text{2) Apply the product property.}\\
&\sqrt[5]{640}\ & \ &\text{3) Multiply.}\\
&\sqrt[5]{32\cdot20}\ & \ &\text{4) Find the largest perfect fifth power factor.}\\
&\sqrt[5]{32}\cdot\sqrt[5]{20}\ & \ &\text{5) Apply the product property.}\\
&2\sqrt[5]{20}\ & \ &\text{6) Simplify the fifth root of 32.}\end{align}\)
&\sqrt[5]{10\cdot64}\ & \ &\text{2) Apply the product property.}\\
&\sqrt[5]{640}\ & \ &\text{3) Multiply.}\\
&\sqrt[5]{32\cdot20}\ & \ &\text{4) Find the largest perfect fifth power factor.}\\
&\sqrt[5]{32}\cdot\sqrt[5]{20}\ & \ &\text{5) Apply the product property.}\\
&2\sqrt[5]{20}\ & \ &\text{6) Simplify the fifth root of 32.}\end{align}\)
Addition and subtraction of \(n^{th}\) roots
To add or subtract \(n^{th}\) roots, the radicands must be equal and the index must be the same (the index is the number in the corner of the radical sign). Do not EVER add or subtract the numbers under the radical sign!
To add or subtract \(n^{th}\) roots:
- Simplify each radical, if necessary.
- If the radicands are equal, add or subtract the coefficients (the numbers outside of the radical). It’s like combining like terms but instead of having a variable, you have an \(n^{th}\) root.
Example 4: Simplify \(4\sqrt[3]{40}+3\sqrt[3]{135}\).
Example 5: Simplify \(-5\sqrt[4]{243}+2\sqrt[4]{48}+4\sqrt[3]{40}\).
Finally, we can simplify radicals that have variables as part of the radicand. The process is similar to simplifying any \(n^{th}\) root--you will simplify the numerical portion the same way as shown above (either by prime factoring or using the largest \(n^{th}\) power factor) AND you will also simplify the \(n^{th}\) root of each variable raised to a power using the multiplication of like bases rules of exponents (\(\sqrt[n]{x^n}=x\) if \(n\) is odd and \(\sqrt[n]{x^n}=\left|x\right|\) if \(n\) is even) as shown in the video examples below.
There is one extra step that is required when you taken an even root (n = 2, 4, 6, …) of variables. When working with real numbers, it is not possible to take an even root of a negative value. Therefore, to guarantee that the variables on the outside of the radical are positive, you must enclose all variables raised to an odd power inside the absolute value symbol. It is not necessary to use absolute value if the variable is raised to an even power, because any base raised to an even power results in a positive value.
For example, \(\sqrt[4]{x^5}=\left|x\right|\sqrt[4]{x}\) and \(\sqrt[4]{x^{13}}=\left|x^3\right|\sqrt[4]{x}\) but \(\sqrt[4]{x^9}=x^2\cdot\sqrt[4]{x}\).
Example 6: Simplify \(\sqrt[4]{48x^5y^4z^9}\).
Example 7: Simplify \(\sqrt[5]{32x^7y^4z^8}\).
Example 8: Simplify \(z\sqrt[3]{24x^7y^4z^2}+xy\sqrt[3]{3x^4yz^5}\).