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nth Roots

Manipulate 3rd and greater roots

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Defining nth Roots

The volume of a cube is found to be \begin{align*}343s^7\end{align*}. What is the length of each side of the cube?

nth Roots

So far, we have seen exponents with integers and the square root. In this concept, we will link roots and exponents. First, let’s define additional roots. Just like the square and the square root are inverses of each other, the inverse of a cube is the cubed root. The inverse of the fourth power is the fourth root.

\begin{align*}\sqrt[3]{27}=\sqrt[3]{3^3}=3, \sqrt[5]{32}=\sqrt[5]{2^5}=2\end{align*}

The \begin{align*}n^{th}\end{align*} root of a number, \begin{align*}x^n\end{align*}, is \begin{align*}x, \sqrt[n]{x^n}=x\end{align*}. And, just like simplifying square roots, we can simplify \begin{align*}n^{th}\end{align*} roots.

Let's find \begin{align*}\sqrt [6]{729}\end{align*}.

To simplify a number to the sixth root, there must be 6 of the same factor to pull out of the root.

\begin{align*}729=3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3=3^6\end{align*}

Therefore, \begin{align*}\sqrt[6]{729}=\sqrt[6]{3^6}=3\end{align*}. The sixth root and the sixth power cancel each other out. We say that 3 is the sixth root of 729.

From this problem, we can see that it does not matter where the exponent is placed, it will always cancel out with the root.

\begin{align*}\sqrt[6]{3^6}&=\sqrt[6]{3}^6 \ or \ \left(\sqrt[6]{3} \right)^6\\ \sqrt[6]{729}&=\left(1.2009 \ldots\right)^6\\ 3&=3\end{align*}

So, it does not matter if you evaluate the root first or the exponent.

The \begin{align*}n^{th}\end{align*} Root Theorem: For any real number \begin{align*}a\end{align*}, root \begin{align*}n\end{align*}, and exponent \begin{align*}m\end{align*}, the following is always true: \begin{align*}\sqrt[n]{a^m}=\sqrt[n]{a}^m=\left(\sqrt[n]{a}\right)^m\end{align*}.

Now, let's evaluate the following expressions without a calculator.

  1. \begin{align*}\sqrt[5]{32^3}\end{align*}

If you solve this problem as written, you would first find \begin{align*}32^3\end{align*} and then apply the \begin{align*}5^{th}\end{align*} root.


However, this would be very difficult to do without a calculator. This is an example where it would be easier to apply the root and then the exponent. Let’s rewrite the expression and solve.


  1. \begin{align*}\sqrt{16}^3\end{align*}

This problem does not need to be rewritten. \begin{align*}\sqrt{16}=4\end{align*} and then \begin{align*}4^3 = 64\end{align*}.

Finally, let's simplify the following.

  1. \begin{align*}\sqrt[4]{64}\end{align*}

To simplify the fourth root of a number, there must be 4 of the same factor to pull it out of the root. Let’s write the prime factorization of 64 and simplify.

\begin{align*}\sqrt[4]{64}=\sqrt[{\color{red}4}]{{\color{red}2 \cdot 2\cdot 2\cdot 2}\cdot 2\cdot 2}=2\sqrt[4]{4}\end{align*}

Notice that there are 6 2’s in 64. We can pull out 4 of them and 2 2’s are left under the radical.

  1. \begin{align*}\sqrt[3]{\frac{54x^3}{125y^5}}\end{align*}

Just like simplifying fractions with square roots, we can separate the numerator and denominator.

\begin{align*}\sqrt[3]{\frac{{54x^3}}{{125y^5}}}= \frac{\sqrt[3]{54x^3}}{\sqrt[3]{125y^5}}=\frac{\sqrt[{\color{red}3}]{2 \cdot{\color{red}3\cdot 3\cdot 3}\cdot {\color{red}x^3}}}{\sqrt[{\color{blue}3}]{{\color{blue}5\cdot 5\cdot 5\cdot y^3}\cdot y^2}}=\frac{3x\sqrt[3]{2}}{5y\sqrt[3]{y^2}}\end{align*}

Notice that because the \begin{align*}x\end{align*} is cubed, the cube and cubed root cancel each other out. With the \begin{align*}y\end{align*}-term, there were five, so three cancel out with the root, but two are still left under radical.


Example 1

Earlier, you were asked to find the length of each side of the cube. 

Recall that the volume of a cube is \begin{align*}V = s^3\end{align*}, where s is the length of each side. So to find the side length, take the cube root of \begin{align*}343z^7\end{align*}.

First, you can separate this number into two different roots, \begin{align*}\sqrt[3]{343} \cdot \sqrt[3]{z^7}\end{align*}. Now, simplify each root.

\begin{align*}\sqrt[3]{343}\cdot \sqrt[3]{z^7} = \sqrt[3]{7^3}\cdot \sqrt[3]{z^3\cdot z^3 \cdot z}= 7z^2 \sqrt[3]{z}\end{align*}

Therefore, the length of the cube's side is \begin{align*}7z^2 \sqrt[3]{z}\end{align*}.

Simplify each expression below, without a calculator.

Example 2


First, you can separate this number into two different roots, \begin{align*}\sqrt[4]{625} \cdot \sqrt[4]{z^8}\end{align*}. Now, simplify each root.

\begin{align*}\sqrt[4]{625}\cdot \sqrt[4]{z^8} = \sqrt[4]{5^4}\cdot \sqrt[4]{z^4\cdot z^4}= 5z^2\end{align*}

When looking at the \begin{align*}z^8\end{align*}, think about how many \begin{align*}z^4\end{align*} you can even pull out of the fourth root. The answer is 2, or a \begin{align*}z^2\end{align*}, outside of the radical.

Example 3


\begin{align*}32 = 2^5\end{align*}, which means there are not 7 2's that can be pulled out of the radical. Same with the \begin{align*}x^5\end{align*} and the \begin{align*}y\end{align*}. Therefore, you cannot simplify the expression any further.

Example 4


Write out 9216 in the prime factorization and place factors into groups of 5.

\begin{align*}\sqrt[5]{9216}&=\sqrt[5]{\boxed{2 \cdot 2\cdot 2\cdot 2\cdot 2}\cdot \boxed{2\cdot 2 \cdot 2\cdot 2\cdot 2} \cdot 3\cdot 3}\\ &=\sqrt[5]{2^5\cdot 2^5 \cdot 3^2}\\ &=2\cdot 2 \sqrt[5]{3^2}\\ &=4\sqrt[5]{9}\end{align*}

Example 5


Reduce the fraction, separate the numerator and denominator and simplify.

\begin{align*}\sqrt[3]{\frac{40}{175}}=\sqrt[3]{\frac{8}{35}}=\frac{\sqrt[3]{2^3}}{\sqrt[3]{35}}={\color{red}{\frac{2}{\sqrt[3]{35}}}\cdot} {\color{red}{\frac{\sqrt[3]{35^2}}{\sqrt[3]{35^2}}}}=\frac{2\sqrt[3]{1225}}{35}\end{align*}

In the red step, we rationalized the denominator by multiplying the top and bottom by \begin{align*}\sqrt[3]{35^2}\end{align*}, so that the denominator would be \begin{align*}\sqrt[3]{35^3}\end{align*} or just 35. Be careful when rationalizing the denominator with higher roots!


Reduce the following radical expressions.

  1. \begin{align*}\sqrt[3]{81}\end{align*}
  2. \begin{align*}\sqrt[4]{625}^3\end{align*}
  3. \begin{align*}\sqrt{9^5}\end{align*}
  4. \begin{align*}\sqrt[5]{128}\end{align*}
  5. \begin{align*}\sqrt{\sqrt{10000}}\end{align*}
  6. \begin{align*}\sqrt{\frac{25}{8}}^4\end{align*}
  7. \begin{align*}\sqrt[6]{64}^5\end{align*}
  8. \begin{align*}\sqrt[3]{\frac{8}{81}}^2\end{align*}
  9. \begin{align*}\sqrt[4]{\frac{243}{16}}\end{align*}
  10. \begin{align*}\sqrt[3]{24x^5}\end{align*}
  11. \begin{align*}\sqrt[4]{48x^7y^{13}}\end{align*}
  12. \begin{align*}\sqrt[5]{\frac{160x^8}{y^7}}\end{align*}
  13. \begin{align*}\sqrt[3]{1000x^6}^2\end{align*}
  14. \begin{align*}\sqrt[4]{\frac{162x^5}{y^3z^{10}}}\end{align*}
  15. \begin{align*}\sqrt{40x^3y^4}^3\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 7.1. 

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n^{th} root The n^{th} root of a number, x^n, is x. Therefore: \sqrt[n]{x^n}=x.
nth root The n^{th} root of a number, x^n, is x. Therefore: \sqrt[n]{x^n}=x.

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