Problem 84 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 81-84: Simplify the expression, assuming \(x\) and \(y\) may be negative. $$ \sqrt[4]{(x+2)^{12} y^{4}} $$

Step-by-Step Solution

Answer

The simplified form is \( (x+2)^3 y \).

Step 1: Recognize the Expression

We are given the expression \( \sqrt[4]{(x+2)^{12} y^{4}} \). This indicates that we need to determine the 4th root of \((x+2)^{12} y^{4}\) and simplify the expression.

Step 2: Separate into Powers of 4

Recognize that each term under the root can be expressed in terms of a power of 4. The expression \( (x+2)^{12} y^{4} \) can be separated into \( ((x+2)^3)^{4} \times y^4 \).

Step 3: Apply the 4th Root

By properties of exponents, take the 4th root of each separate term: \( \sqrt[4]{((x+2)^3)^4} \times \sqrt[4]{y^4} \). This simplifies to \((x+2)^3 \times y\), since \(\sqrt[4]{a^4} = a\).

Step 4: Combine the Simplified Terms

Combine the simplified results to conclude that \( \sqrt[4]{(x+2)^{12} y^{4}} = (x+2)^3 y \).

Key Concepts

Radicals

In mathematics, radicals are symbols used to represent roots of numbers or expressions. A common radical is the square root, denoted as \( \sqrt{} \), but radicals can represent any nth root like cube roots (\( \sqrt[3]{} \)) or 4th roots, as seen in our exercise (\( \sqrt[4]{} \)).
Radicals follow certain rules that help in simplifying expressions:

• You can separate a radical into the root of each term multiplied inside, like \( \sqrt[4]{ab} = \sqrt[4]{a} \times \sqrt[4]{b} \).
• Taking a root reverses exponentiation. For instance, \( \sqrt[4]{a^4} = a \).

Understanding these rules allows us to simplify complex expressions involving roots, making subsequent calculations more manageable.

Exponents

Exponents are mathematical symbols that indicate how many times a number, known as the base, is multiplied by itself. They serve to simplify repeated multiplication. For example, \((x+2)^{12}\) implies multiplying \((x+2)\) by itself 12 times.
Key properties of exponents useful in simplifying expressions include:

• Multiplying powers: When multiplying terms with the same base, you add the exponents, such as \(a^m \times a^n = a^{m+n}\).
• Power of a power: Taking an exponent to another power requires multiplying the exponents, indicated by \((a^m)^n = a^{m \cdot n}\).
• Roots as fractional exponents: \(\sqrt[n]{a} = a^{1/n}\).

These rules are crucial in the step-by-step simplification of expressions containing exponents.

Simplifying Expressions

Simplifying expressions means reducing them to their most compact form, while ensuring they remain mathematically equivalent. This often involves combining like terms, using laws of exponents and radicals, and reducing any complex multiplications or factorizations.
In our original exercise, simplification consisted of:

• Identifying powers and grouping them based on the radicals involved.
• Recognizing and applying the rules of exponents to break down terms into simpler forms.
• Utilizing the property of radicals to individually simplify each factor within the larger expression, as per the expression's roots.

The aim of simplification is to make expressions easier to read, compare, and work with in further mathematical operations.