Problem 81 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 81-84: Simplify the expression, assuming \(x\) and \(y\) may be negative. $$ \sqrt{x^{6} y^{4}} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to simplify the given expression \( \sqrt{x^6 y^4} \). This expression involves square roots and involves the powers of variables, \(x\) and \(y\). We should look to simplify this under the assumption that both variables could be negative.

Step 2: Breaking Down the Expression

The expression inside the square root is \(x^6 y^4\). We can write this as the product of two separate terms: \((x^6)\cdot(y^4)\). We know that the square root of a product is the product of the square roots: \(\sqrt{x^6 y^4} = \sqrt{x^6}\cdot\sqrt{y^4}\).

Step 3: Simplifying Each Part

First, simplify \(\sqrt{x^6}\). The square root of \(x^6\) is \(x^{6/2} = x^3\). Since \(x\) might be negative, we should ensure this is a non-negative result; hence it's \(|x^3|\).Next, simplify \(\sqrt{y^4}\). The square root of \(y^4\) is \(y^{4/2} = y^2\). Since \(y^2\) is always non-negative, no absolute value is needed here.

Step 4: Combining the Simplified Results

Combine the results from the previous step: \(|x^3| \cdot y^2\). This is the simplified form of the original expression, ensuring correctness for negative values of \(x\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is like tidying up your math. The goal is to make expressions as neat and straightforward as possible.

• Look for patterns or rules, like factoring or distributing.
• Break down complex terms into simple ones.

This can involve combining like terms or resolving operations, such as exponents. In our exercise, we take the square root of a product and break it down. This involves simplifying each separate part before combining them back together. It helps to remember that every operation follows the same rules, regardless of how complex an expression looks at first glance.
Once simplified, the expression is often much easier to work with, especially when variables can take on any values, like being positive or negative.

Square Roots

A square root essentially "undoes" a square, searching for a number that multiplies by itself to give the original value.

• The square root of 9 is 3, because 3 times 3 is 9.
• Symbolically, we use \( \sqrt{9} \) to denote the square root of 9.

When dealing with square roots of variables, you handle the operations similar to numbers. For example, \( \sqrt{x^6} \) involves finding the power that combines to build back to \( x^6 \).
We achieve this by using the property of exponents: the square root of \( x^n \) simplifies to \( x^{n/2} \). Therefore, \( \sqrt{x^6} \) becomes \( x^3 \).
However, since variables could be negative in real-world problems, we use absolute values to ensure non-negative results, thus giving us \( |x^3| \). This helps maintain correctness, especially in algebraic expressions.

Powers of Variables

Powers, or exponents, are a way to express repeated multiplication of the same number or variable. When you see \( x^6 \), it signifies that \( x \) is multiplied by itself six times.

• \( x^n \) indicates repeated multiplication of \( x \) by itself \( n \) times.
• The laws of exponents help simplify expressions, such as \((x^a)^b = x^{a \times b}\).

In our example, when simplifying the square root of \( x^6 \), we use the principle that \( \sqrt{x^n} = x^{n/2} \). This rule allows us to express \( x^6 \) as \( x^{6/2} = x^3 \).
Using powers efficiently helps in managing large computations and breaking them down into simpler steps.
Always keep in mind the rules of negative bases, since negative numbers raised to even powers yield positive results, simplifying absolute values further.

Absolute Value

The concept of absolute value is crucial when dealing with algebraic expressions involving variables that can be negative. The absolute value of a number or expression is its distance from zero on a number line.

• The absolute value of -3 is 3, expressed as \(|-3| = 3\).
• For any variable \( x \), \(|x|\) ensures a non-negative result, regardless of \( x \)'s original sign.

In algebra, when variables are involved in powers, like \( |x^3| \), it ensures that even if \( x \) is negative, the expression remains non-negative.
Absolute values are especially important when simplifying square roots of variables, as they automatically adjust to the non-negative square values. This feature maintains the mathematical integrity of expressions under all conditions.