Question

\(\frac{\sqrt[3]{5}}{\sqrt[3]{4}}\)

Short answer

The simplified version of the given expression \(\frac{\sqrt[3]{5}}{\sqrt[3]{4}}\) is \(\sqrt[3]{1.25}\).

Step-by-step solution

Identify the given expression

The given expression is \(\frac{\sqrt[3]{5}}{\sqrt[3]{4}}\). The idea here is to simplify this expression. Since both terms are under the cubic root and in the format of \(\sqrt[3]{a}/\sqrt[3]{b}\), we could apply the quotient rule of radicals here. The quotient rule of radicals states that \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\), where a and b are positive real numbers and n is an integer.

Recall and apply the quotient rule of radicals

The quotient rule of radicals can be applied to our expression. This leads us to \(\sqrt[3]{\frac{5}{4}}\). Now, the expression is in a simpler format.

Convert the fraction under the cube root into decimal

The fraction 5/4 can be converted into decimal as \(\sqrt[3]{1.25}\). Now the expression should be simpler to read and understand.

Key concepts

Cube Roots

Cube roots are a specific type of radical expression that involves finding a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because 2 × 2 × 2 = 8. In mathematical notation, cube roots are represented as \(\sqrt[3]{x}\), where \(x\) is the number we want to find the cube root of and 3 is the index indicating it's a cube root.
Cube roots are important in many areas of math and science because they allow us to simplify expressions and solve equations involving cubic terms. Understanding how to work with cube roots will enable you to simplify complex equations and expressions in a more straightforward manner.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This often means removing unnecessary terms, combining like terms, or using mathematical rules to condense the expression. The goal is to make the expression as easy to work with as possible.
In the context of cube roots and other radicals, simplification can involve applying rules such as the quotient rule of radicals. This rule allows us to combine or separate radical expressions in a way that may make them easier to understand or calculate.

• Start by identifying parts of the expression that can be simplified. For cube roots, look for numbers or variables that can be combined or whose cube roots can be easily calculated.
• Use mathematical rules, such as distributing, factoring, and canceling terms, to assist in the simplification process.
• Simplifying expressions often requires practice and familiarity with algebraic rules and properties.

This makes it easier to solve the problem or understand the relationship between different elements of an expression.

Radical Expressions

Radical expressions are mathematical expressions that include a radical symbol, \(\sqrt{}\), which signifies taking a root of a number. The type of root is indicated by a small number, called the index, placed just above and to the left of the radical symbol. If no index is shown, it is assumed to be a square root.
Handling radical expressions correctly is crucial because they appear in many areas of mathematics, such as solving equations, integrating functions, and even in geometry when dealing with lengths and distances. The rules for manipulating radical expressions, such as the quotient rule and product rule, are fundamental techniques for working with these expressions.
By applying these rules, you can manage expressions efficiently, helping to achieve a precise and simplified form. This process also aids in systematically solving more complex mathematical problems.