Problem 38 - Algebra for College Students

Question

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \((5 x+y)^{\frac{1}{3}}\)

Step-by-Step Solution

Answer

The expression \((5x + y)^{\frac{1}{3}}\) in radical form is \(\sqrt[3]{5x + y}\).

Step 1: Identify the Expression

Our task is to rewrite the expression \((5x + y)^{\frac{1}{3}}\) in radical form.

Step 2: Understand the Exponent Notation

The expression is in fractional exponent notation, specifically of the form \(a^{\frac{m}{n}}\). This corresponds to the radical form \(\sqrt[n]{a^m}\).

Step 3: Apply to the Given Expression

For the expression \((5x + y)^{\frac{1}{3}}\), we have \(m = 1\) and \(n = 3\). This translates to \( \sqrt[3]{(5x + y)^1}\).

Step 4: Simplify the Radical Expression

Since the power of the radicand is 1, it simplifies directly to \( \sqrt[3]{5x + y} \).

Key Concepts

Fractional Exponents

Fractional exponents are a unique way to express powers and roots in one compact form. In mathematics, a fractional exponent like \(a^{\frac{m}{n}}\) combines both the power and the root of a number. The numerator \(m\) represents the power, while the denominator \(n\) signifies the root. For example, \(x^{\frac{2}{3}}\) translates to \(\sqrt[3]{x^2}\), indicating the cube root of \(x\) squared.
Using fractional exponents helps streamline complex calculations by merging two operations - exponentiation and taking roots - into one expression. This dual representation supports solving equations more efficiently as it directly connects powers and roots. Learning to handle fractional exponents is essential for working with advanced algebraic expressions, as many transformations and simplifications involve these forms.
In our exercise, converting \((5x + y)^{\frac{1}{3}}\) involves identifying the base \((5x + y)\), taking it to the power of 1, and then applying the cube root.

Radicals

Radicals are symbols that denote the root of a number. When you see a radical form, like \(\sqrt[n]{a}\), it specifies which root to take. The symbol \(\sqrt{}\) is called a radical, and the number inside, \(a\), is known as the radicand. The index \(n\) suggests the degree of the root. When \(n\) is 2, the radical \(\sqrt{a}\) describes the square root, and for \(n = 3\), it describes the cube root, as in the current exercise \(\sqrt[3]{5x + y}\).
To approach radicals correctly:

• Determine the index \(n\)
• Identify the radicand
• Simplify if possible, such as combining like terms or reducing powers

These steps ensure that the expression is simplified and clearly understood. It's crucial to handle radicals correctly, especially when they appear in algebraic contexts, as they are foundational in developing further skills in calculus and advanced mathematics. In practical terms, dealing with a radical simplifies complex expressions by breaking them down to their root components.
In our solution, converting from \((5x + y)^{\frac{1}{3}}\) to \(\sqrt[3]{5x + y}\) seamlessly transforms a fractional exponent into the language of radicals.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra, serving as a language that encodes mathematical relationships and structures. An expression like \(5x + y\) consists of terms where \(5x\) and \(y\) are combined using addition. Within these expressions, operations are applied following the rules of mathematics such as distributive, associative, and commutative properties.
Key features of algebraic expressions include:

• Variables (e.g., \(x, y\)) representing unknown values
• Constants (e.g., numbers like 5)
• Operations (+, -, *, /) connecting the terms

In sophisticated problems, understanding these expressions' structure helps in finding solutions, as they can be manipulated using algebraic techniques like factoring and expanding.
In our example, \(5x + y\) is a simple two-term expression. When raised to the fractional power of \(\frac{1}{3}\), it becomes part of a more complex expression, which is then simplified to a radical form. Mastery of algebraic expressions ensures that students can smoothly transition between these different forms, effectively harnessing the power of algebra in their calculations.