Problem 57 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\frac{\sqrt[3]{3 y}}{\sqrt[3]{16 x^{4}}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt[3]{3y}}{\sqrt[3]{16x^4}} = \frac{\sqrt[3]{3} \cdot \sqrt[3]{y}}{\sqrt[3]{16} \cdot \sqrt[3]{x^4}}\)

Step 1: Simplify the denominator

The denominator is \( \sqrt[3]{16x^4} \). We can express 16 as \( 2^4 \), so the expression becomes \( \sqrt[3]{2^4x^4} = \sqrt[3]{(2x)^4} \). This simplifies to \((2x)^{4/3}\).

Step 2: Write the expression as a quotient

Rewrite the entire expression as a single fraction with exponents as follows:\[ \frac{(3y)^{1/3}}{(2x)^{4/3}} \]

Step 3: Simplify by subtracting exponents

Use the laws of exponents to rewrite the expression as \((3y)^{1/3} \times (2x)^{-4/3}\). This becomes:\[ 3^{1/3} y^{1/3} \times 2^{-4/3} x^{-4/3} \]

Step 4: Simplify each component

Combine the exponents to form a simpler expression:\[ \frac{3^{1/3}y^{1/3}}{2^{4/3}x^{4/3}} \]This expression can be further expressed as\[ \frac{\sqrt[3]{3y}}{\sqrt[3]{16x^{4}}} = \frac{\sqrt[3]{3} \cdot \sqrt[3]{y}}{\sqrt[3]{16} \cdot \sqrt[3]{x^4}} \]

Key Concepts

Radicals

Radicals are a fundamental part of algebra and often come up when you work with functions and equations. In simple terms, a radical is an expression that includes a root, such as square roots, cube roots, or other higher-order roots. A common form of radicals is

• Square roots (e.g., \(\sqrt{a}\) for the square root of \(a\))
• Cube roots (e.g., \(\sqrt[3]{b}\) for the cube root of \(b\))

Each number under the root sign is called the radicand (e.g., in \(\sqrt[3]{16x^{4}}\), \(16x^{4}\) is the radicand).

To express a radical in simplest form, you'll look for perfect powers that can be factored out of the radicand and simplified. For example, with cube roots, you would search for terms in the radicand that are perfect cubes.

The original problem involves cube roots, and simplifying these involves factoring powers of the terms involved and adjusting them accordingly with the use of exponents. Understanding radicals deeply is essential for managing algebra problems efficiently.

Exponents

Exponents are a way to express repeated multiplication of a number or a variable. For example, \(x^3\) means \(x\) multiplied by itself three times.

There are some basic rules of exponents that are super helpful:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(a^m / a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m\cdot n}\)

Additionally, negative exponents such as \(x^{-n}\) mean \(1/x^n\).

In the given exercise, exponent rules are key to simplifying the radical expression. It involves transforming radicals into fractional exponents and applying these laws to consolidate and simplify the components of the algebraic expressions involved in the solution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can be as simple as \(x + 2\) or more complex like the given expression in the problem.

An understanding of how to manipulate these expressions is crucial given that they often need simplification or reorganization for easier solving. Simplifying involves:

• Using operations like addition or multiplication
• Applying properties of numbers such as combining like terms
• Employing rules for algebraic functions, such as the order of operations

The expression in square root form is broken down and reorganized using exponent laws in steps. Woodwork like this involves altering the arrangement of variables and constants to arrive at the simplest radical form.

Mastery of these expressions will assist with solving a variety of equations and understanding deeper algebraic principles, laying a foundation for advancing in mathematical studies.