Problem 34 - Algebra and Trigonometry

Question

\(29-38=\) Simplify the expression. Assume that the letters denote any real numbers. $$ \sqrt{x^{4} y^{4}} $$

Step-by-Step Solution

Answer

\( \sqrt{x^4 y^4} = x^2 y^2 \)

Step 1: Understand the Problem

We have an expression \( \sqrt{x^{4} y^{4}} \) which involves a square root of a product of two powers. Our task is to simplify this expression.

Step 2: Apply the Property of Square Roots

Recall that for any non-negative real numbers \(a\) and \(b\), \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). We will apply this property to the expression: \[ \sqrt{x^4 y^4} = \sqrt{x^4} \cdot \sqrt{y^4} \].

Step 3: Simplify Each Square Root

Next, we simplify each square root separately. Recall that \( \sqrt{a^2} = a \) for any non-negative \(a\):- \( \sqrt{x^4} = x^{4/2} = x^2 \)- \( \sqrt{y^4} = y^{4/2} = y^2 \) Thus, \( \sqrt{x^4} \cdot \sqrt{y^4} = x^2 y^2 \).

Step 4: Combine the Results

Combine the simplified components from Step 3 to find the final simplified expression. Thus, \( \sqrt{x^4 y^4} = x^2 y^2 \).

Key Concepts

Understanding Square Roots

The concept of a square root is fundamental in algebra and mathematics. To "take the square root" of a number or an expression means to find a value that, when multiplied by itself, equals the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\). This operation is typically represented by the radical symbol \(\sqrt{\ }\).

When simplifying expressions under a square root, it's important to note the property that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This allows us to simplify complex expressions, such as products, into separate components. In the given example, \( \sqrt{x^4 y^4} \), you can first apply this rule to split the expression into \( \sqrt{x^4} \) and \( \sqrt{y^4} \).

It's also crucial to remember that square roots are generally only defined for non-negative numbers in the set of real numbers. This ensures that the result is itself a non-negative real number.

Exploring Exponents

Exponents are a way to represent repeated multiplication of a number by itself. In general, \(x^n\) means that x is multiplied by itself n times. In our expression, even though we start with \(x^4\) and \(y^4\), the process of simplifying these involves understanding their root. Remember, \(x^{4/2} = x^2\) simplifies the exponent by dividing it by 2, which corresponds to taking the square root.

The laws of exponents, such as \((x^a)^b = x^{a \cdot b}\), help simplify expressions as well. They demonstrate how multiplying powers can be reduced to simpler exponents, like in the step \(x^4\) becomes \(x^{2}\) after taking its square root.

When dealing with expressions involving exponents, multiplication within these expressions can often be handled effectively using these laws, particularly when combined with roots, as seen in the simplified result of \(x^2 y^2\), representing the square of each base.

Understanding Real Numbers

Real numbers encompass a broad range of values, including all the rational and irrational numbers. They include integers, fractions, and decimals, both terminating and non-terminating. Essentially, real numbers can be found on the number line and are used to measure every conceivable quantity in standard mathematics.

Given this breadth, operations and simplifications involving real numbers are well-defined, allowing us to simplify expressions like \(\sqrt{x^4 y^4}\) with the assurance it remains within the realm of real numbers. It confirms the use of properties and rules like square roots and exponents.

In essence, when simplifying algebraic expressions such as the one provided, understanding that \(x\) and \(y\) represent any real numbers is crucial in ensuring the manipulated expressions follow mathematical conventions and maintain their status as real numbers.