Question

Simplify the expression. $$ \frac{\sqrt[4]{405}}{\sqrt[4]{5}} $$

Short answer

The simplified version of the expression \( \frac{\sqrt[4]{405}}{\sqrt[4]{5}} \) is 3

Step-by-step solution

Identify the Rule

Recognize that the expression is set up as a quotient of two fourth root terms. According to the rule \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \), you can divide the numbers inside the radical first before taking the fourth root.

Divide the Numbers Inside the Radical

Next, divide 405 by 5 inside the radical. \( \sqrt[4]{\frac{405}{5}} = \sqrt[4]{81} \)

Simplify the Fourth Root

Simplify the fourth root of 81. The fourth root of 81 is 3 because \( 3^4 = 81 \). So, \( \sqrt[4]{81} = 3 \)

Key concepts

Fourth Roots

A fourth root of a number is another number which, when raised to the power of four, equals the original number. For instance, in the expression \( \sqrt[4]{81} \), you are asked to find the number that when multiplied by itself four times results in 81. The number that fits this requirement is 3, as \( 3^4 = 81 \).
Understanding fourth roots is similar to understanding square roots, but instead of looking for a square (or a number raised to the power of 2), you’re looking for a fourth power. It’s a crucial concept in mathematics that shows up when dealing with radical expressions, particularly in problems requiring simplification.

Quotients

In mathematics, a quotient is the result of division. When you see the symbol \( \frac{a}{b} \), it means \( a \) is being divided by \( b \). When dealing with radical expressions, like \( \frac{\sqrt[4]{405}}{\sqrt[4]{5}} \), the quotient underlies the process of simplifying expressions.
The rule for dividing two like radicals is important in simplification: \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \).

• This rule allows you to combine the terms inside the radicals first, which can make expressions significantly easier to work with.
• For example, in \( \frac{\sqrt[4]{405}}{\sqrt[4]{5}} \), you can simplify this to \( \sqrt[4]{\frac{405}{5}} \) before finding the fourth root, simplifying computations greatly.

Simplification

Simplification in mathematics is the process of breaking down expressions into simpler or more compact form.

• The goal is often to perform calculations more easily or to gain insight into the problem.
• Specifically, with radicals, simplification involves moving under the radical sign into manageable numbers or reducing complex radicals to basic numerical values.

Take the expression \( \sqrt[4]{\frac{405}{5}} \). By dividing inside the radical, you get \( \sqrt[4]{81} \). This transformation allows you to reduce a cumbersome radical expression, resulting in a simple integer: 3.

Mathematical Expressions

Mathematical expressions consist of numbers, symbols, and operators arranged in a specific manner to define particular value relations or properties. In the context of the given exercise, we work with core components like radicals and quotients to simplify expressions.
A well-simplified mathematical expression is not only easier to read but often provides better insight into its arithmetic structure. For instance, \( \frac{\sqrt[4]{405}}{\sqrt[4]{5}} \) simplifies into an easy-to-understand value of 3.
This simplification process helps you to avoid fractional uncertainty and clearly present results or make further calculations. Such exercises improve your ability to manipulate and understand algebraic or numerical relationships.