Question

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt[4]{\frac{x^{9} y^{7}}{x y^{3}}}$$

Short answer

The simplified expression is \(x^{2} y\).

Step-by-step solution

Simplify the expression inside the radical

First, simplify the expression inside the radical. Divide the numerator and the denominator: \[\frac{x^{9} y^{7}}{x y^{3}} = \frac{x^{9}}{x} \cdot \frac{y^{7}}{y^{3}} = x^{8} \cdot y^{4} \]

Apply the fourth root to the simplified expression

Next, apply the fourth root to both the terms inside the radical: \[\sqrt[4]{x^8 y^4} = \sqrt[4]{x^8} \cdot \sqrt[4]{y^4} \]

Simplify each term

Simplify each term separately: \[\sqrt[4]{x^8} = x^{\frac{8}{4}} = x^{2} \] \[\sqrt[4]{y^4} = y^{\frac{4}{4}} = y\]

Combine the simplified terms

Combine the simplified terms to get the final result: \[x^{2} \cdot y \]

Key concepts

Fourth Root

The fourth root of a number is a value that, when multiplied by itself four times, gives that number. For example, the fourth root of 16 is 2 because \(2 \cdot 2 \cdot 2 \cdot 2 = 16\).
When dealing with variables, the process is similar. If we have \(\sqrt[4]{x^8}\), we need to find a value that, when raised to the power of 4, equals \(x^8\). This can be done by dividing the exponent by 4. So, \(\sqrt[4]{x^8} = x^{\frac{8}{4}} = x^2\).
The fourth root is often used in algebra to simplify radical expressions, making them easier to work with in equations and other mathematical operations.

Exponent Rules

Exponent rules are essential for simplifying expressions with radicals. One fundamental rule is the division of exponents: when you divide powers with the same base, you subtract the exponents. For example, \[\frac{x^a}{x^b} = x^{a-b}\].
Another important rule applies when dealing with roots: the nth root of a power can be expressed as the power divided by n. For example, \[\sqrt[n]{x^m} = x^{\frac{m}{n}}\].
In our exercise, we use these rules to simplify \[\sqrt[4]{\frac{x^{9} y^{7}}{x y^{3}}}\]. First, we simplify the exponents inside the fraction: \[\frac{x^{9}}{x} = x^{9-1} = x^8\] and \[\frac{y^{7}}{y^3} = y^{7-3} = y^4\]. Then, we apply the rule for roots to each term: \[\sqrt[4]{x^8} = x^2\] and \[\sqrt[4]{y^4} = y\].

Simplifying Fractions

Simplifying fractions involves reducing the numerator and denominator to their simplest forms. This often involves canceling out common factors. For example, in the fraction \[\frac{x^{9} y^{7}}{x y^{3}}\], we break it down into two separate fractions: \[\frac{x^{9}}{x} \] and \[\frac{y^{7}}{y^3}\].
The first fraction simplifies by subtracting the exponents: \[\frac{x^{9}}{x} = x^{9-1} = x^8\].
The second fraction also simplifies by subtraction: \[\frac{y^{7}}{y^3} = y^{7-3} = y^4\].
Combining these, we get \[x^8 y^4\]. Simplification is crucial for handling complex algebraic fractions and making calculations easier.

Radical Expressions

Radical expressions involve roots, such as square roots or fourth roots. Simplifying these expressions can make them easier to work with. For example, \[\sqrt[4]{x^8 y^4}\] involves finding the fourth root of each term inside the expression.
We break it down: \[\sqrt[4]{x^8} = x^2\] because \[\left(x^2\right)^4 = x^8\]. Likewise, \[\sqrt[4]{y^4} = y\] because \[\left(y\right)^4 = y^4\].
Combining these simplified terms, we get the final result, \[x^2 y\].
Being able to simplify radical expressions is important for solving algebraic problems efficiently.