Problem 82 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 81-84: Simplify the expression, assuming \(x\) and \(y\) may be negative. $$ \sqrt{x^{4} y^{10}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^2 y^5 \).

Step 1: Break Down the Expression into Powers

The given expression is \( \sqrt{x^4 y^{10}} \). We can break this down into two separate square roots: \( \sqrt{x^4} \) and \( \sqrt{y^{10}} \). This is done using the property of square roots: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).

Step 2: Simplify Each Square Root Separately

Now, simplify each square root separately: \( \sqrt{x^4} = (x^4)^{1/2} \) and \( \sqrt{y^{10}} = (y^{10})^{1/2} \). Using the power of a power property \( (a^m)^n = a^{m \cdot n} \), compute: \((x^4)^{1/2} = x^{4 \cdot 1/2} = x^2\) and \((y^{10})^{1/2} = y^{10 \cdot 1/2} = y^5\).

Step 3: Combine the Simplified Terms

Combine the simplified terms to get the final expression: \( x^2 y^5 \).

Step 4: Consider Signs of Variables and Final Expression

Since \(x\) and \(y\) can be negative, the simplified expression, in terms of absolute values, is \( |x^2| |y^5| \). However, \( x^2 \) is always non-negative regardless of the sign of \(x\), so it can simply be \( x^2 \). The same logic applies to \( y^5 \), which retains the sign of \( y \) as it is raised to an odd power and shouldn't affect the operation outcome that requires simplification only.

Key Concepts

Simplifying Expressions

Simplifying expressions means reducing an expression to its simplest form. The goal is to make it more manageable and easier to understand, without changing its value. This often involves reducing powers or combining like terms. To simplify expressions effectively, follow these tips:

• Identify and combine like terms. Like terms have the same variables raised to the same powers.
• Use properties of operations, such as the distributive property, to combine expressions.
• Utilize algebraic identities to factor or expand expressions as necessary.

In the exercise involving the square root of \(x^4 y^{10}\), the expression was simplified by separately considering each part under the square root. This was done using the property that allows us to break down a product of terms under a square root into the product of separate square roots. Remember, the simpler an expression is, the easier it will be to use it in calculations.

Square Roots

Square roots are the inverse operation of squaring a number. Finding the square root of a number means identifying a value that, when multiplied by itself, gives the original number. There are some essential properties of square roots:

• The square root of a product \(a \times b\) is the product of the square roots of \(a\) and \(b\): \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
• The square root of a square, such as \(\sqrt{x^2}\), gives the absolute value of the original number: \(|x|\).
• For non-negative numbers, \(\sqrt{x}\) is the principal square root, meaning the non-negative root.

In the solved exercise, the square root of \(x^4 y^{10}\) was split into two separate components, \(\sqrt{x^4}\) and \(\sqrt{y^{10}}\). Then, each was simplified into its respective integer power, yielding a cleaner expression.

Power Properties

Understanding power properties is crucial in simplifying expressions that involve exponents. These properties allow us to manipulate and reduce expressions with powers to their simplest form. Here are some key properties:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^m = a^m \times b^m\)

In the example provided, the power of a power property was used to simplify \((x^4)^{1/2}\) and \((y^{10})^{1/2}\). By applying the property \((a^m)^n = a^{m \cdot n}\), the expression was reduced to \(x^2\) and \(y^5\). This makes managing and calculating expressions involving exponents easier, by efficiently handling their powers.