Problem 16 - Algebra 2

Question

Simplify. Assume that all variables are positive. $$ \sqrt[4]{64 x^{3} y^{6}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt[4]{64x^3y^6}\) is \(2\sqrt{2}x^{3/4}y\sqrt{y}\).

Step 1: Break down constant and variables

The given expression is \(\sqrt[4]{64x^3y^6}\). The constant 64 can be broken down as \(2^6\) since 2 to the power six equals 64. The variables with exponents x and y can be rewritten as \(x^{4*(3/4)}\) and \(y^{4*(6/4)}\). So, the given expression becomes \(\sqrt[4]{2^6x^{4*(3/4)}y^{4*(6/4)}}\).

Step 2: Extract fourth root

Having broken down the constant and the variables with exponents, apply the root to each term. The fourth root of \(2^6\) is \(2^(6/4)\). The fourth root of \(x^{4*(3/4)}\) is \(x^(3/4)\). And, the fourth root of \(y^{4*(6/4)}\) is \(y^(6/4)\). Hence, the expression becomes \(2^{6/4}x^{3/4}y^{6/4}\).

Step 3: Simplify the expression

Simplify the exponents. \(2^{6/4} = 2^{1.5} = 2\sqrt{2}\). \(x^{3/4}\) is simplified as \(x^{3/4}\). \(y^{6/4} = y^{1.5} = y\sqrt{y}\). Therefore, the simplified form of the original fourth root expression is \(2\sqrt{2}x^{3/4}y\sqrt{y}\).

Key Concepts

Simplifying Radicals

Simplifying radicals involves breaking down expressions under a root. This makes them easier to understand and work with, especially when solving equations or performing algebraic operations. To simplify radicals, you should:

• Factor the number or expression under the radical sign.
• Identify and extract individual components from under the root.
• Simplify the expression outside the radical as much as possible.

This step-by-step approach makes handling radical expressions more manageable. For the given exercise, decomposing numbers and variables into their prime factors aids in the simplification process. This reveals simpler forms by applying the rules of exponents and properties of radicals.

Fourth Roots

The concept of fourth roots is similar to square roots, except that you look for a number which, when multiplied by itself four times, gives the original number. Mathematically, the fourth root of an expression like \(\sqrt[4]{a}\) means finding \(a^{1/4}\).

Fourth roots can also apply to entire algebraic expressions, not just numbers. The process involves:

• Breaking down the expression into parts.
• Applying the fourth root to each part.
• Simplifying the results wherever possible.

Understanding this concept is crucial because it allows you to work with more complex expressions by simplifying them into more manageable forms.

Exponents

Exponents are a way to express repeated multiplication of the same number. For instance, \(2^6\) means multiplying 2 by itself six times. Exponents have their own set of rules, such as:

• The product of powers rule: \(a^m \times a^n = a^{m+n}\).
• The power of a power rule: \( (a^m)^n = a^{m \times n} \).
• The power of a product rule: \( (ab)^n = a^n \times b^n \).

In the context of the exercise, the concept of exponents is used to simplify expressions under a radical. Understanding how to manipulate exponents with these rules makes it possible to break down and simplify complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). They allow us to represent real-world problems in mathematical form. Working with algebraic expressions often requires:

• Recognizing and combining like terms.
• Using the distributive property to simplify expressions.
• Understanding and applying the laws of exponents.

In this simplification process, you saw the use of algebraic expressions involving variables with powers. By rewriting and simplifying these terms according to the rules of exponents and radicals, you get a form that is easier to handle and comprehend.