Problem 43 - Algebra 2

Question

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{5+\sqrt[4]{x}}{\sqrt[4]{x}} $$

Step-by-Step Solution

Answer

\(\frac{5\sqrt[4]{x}+\sqrt{x}}{x}\)

Step 1: Multiply numerator and denominator

Multiply both the numerator and the denominator by the same value that clears the denominator of its root. In this case, it is \(\sqrt[4]{x}\), because \((\sqrt[4]{x}) \times (\sqrt[4]{x}) = x\). So rewrite the fraction as \( (\frac{5+\sqrt[4]{x}}{\sqrt[4]{x}}) \times (\frac{\sqrt[4]{x}}{\sqrt[4]{x}})\).

Step 2: Simplify the numerator and the denominator

Solve the multiplication in both the numerator and the denominator: \( \frac{5\sqrt[4]{x}+\sqrt{x}}{x} \)

Step 3: Leave final answer

After simplifying the numerator and the denominator, the final fraction should be \(\frac{5\sqrt[4]{x}+\sqrt{x}}{x}\)

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions makes math problems easier to work with. It involves combining like terms and reducing expressions to their simplest form. The idea is to remove any complexity by eliminating unnecessary factors or terms. For example, when simplifying a fraction like \( \frac{5+\sqrt[4]{x}}{\sqrt[4]{x}} \), we look for ways to make the expression cleaner and more manageable.

• Identify indecomposable parts and objects.
• Look for common factors in terms.
• Apply arithmetic operations to combine or reduce terms.

In the given problem, we aim to get rid of the fourth root in the denominator because it complicates our calculations. We multiply by \( \sqrt[4]{x} \) to eliminate that root and simplify the expression further. This reduces the denominator complexity, making the math simpler to handle.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are algebraic expressions rather than simple numbers. Simplifying these requires a specific approach: First, recognize that rational expressions behave like regular fractions. The rules for adding, subtracting, multiplying, or dividing fractions apply here as well.

• Identify when you can apply factor cancellation.
• Ensure denominators are not zero; find any restrictions on variable values.
• Apply multiplication to clear radicals when needed.

In our case, the original expression \( \frac{5+\sqrt[4]{x}}{\sqrt[4]{x}} \) is a rational expression with a radical in the denominator. By applying multiplication that mirrors the denominator, the radical is cleared effectively, transforming it into a simpler form.

Radicals in Denominators

Dealing with radicals in denominators can often make calculations complex and messy. Rationalizing such expressions is a common technique to remove radicals from the bottom of a fraction, which helps to standardize the calculations. The basic approach involves multiplying by a form of 1 that includes the radical, helping to eliminate it through squaring.

• Multiply by the conjugate if dealing with square roots.
• Use appropriate powers to neutralize the radical.
• Re-write the expression to make it compatible with standard formats.

When rationalizing \( \frac{5+\sqrt[4]{x}}{\sqrt[4]{x}} \), multiplying both the numerator and denominator by \( \sqrt[4]{x} \) aids in eliminating the \( \sqrt[4]{x} \) in the denominator due to the properties of exponents, yielding a more standard form of expression.