Problem 61 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{\sqrt{16 x^{4} y^{4}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2xy\).

Step 1: Identify the innermost square root

Before tackling the entire expression, simplify the innermost square root first. That's \(\sqrt{16 x^{4} y^{4}}\). Here, one can apply the properties of roots, specifically that \(\sqrt{a b}=\sqrt{a} \sqrt{b}\) for all positive \(a\), \(b\).

Step 2: Break down the inner square root

Applying the property, this expression simplifies to \(\sqrt{16} \sqrt{x^{4}} \sqrt{y^{4}}\). Here, \(\sqrt{16} = 4\), and since \(x\) and \(y\) are positive, \(\sqrt{x^4} = x^2\) and \(\sqrt{y^4} = y^2\). So, we have \(4x^2y^2\).

Step 3: Simplify the entire square root

Substitute the simplified expression from step 2 back into the whole expression to get \(\sqrt{4x^2y^2}\). Using the square root property again, this breaks down to \(2xy\).

Key Concepts

Square Root Simplification

Let's begin with the concept of simplifying square roots, which is essential when working with expressions like \(\sqrt{\sqrt{16x^4y^4}}\). Simplifying a square root involves expressing the number or expression under the root sign in its simplest form, making it easier to understand and work with.

In our example, we start with \(\sqrt{16x^4y^4}\), which involves finding the square roots of each factor separately. This is made possible by the property \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\). You can split the expression into separate square roots:

• \(\sqrt{16}\)
• \(\sqrt{x^4}\)
• \(\sqrt{y^4}\)

Once separated, each component can be individually simplified to:

• \(\sqrt{16} = 4\)
• \(\sqrt{x^4} = x^2\)
• \(\sqrt{y^4} = y^2\)

The result from these simplifications is combined to form \(4x^2y^2\). This highlights how breaking down and simplifying is a powerful method for handling square roots.

Properties of Exponents

Understanding the properties of exponents is crucial for both simplifying expressions and rationalizing denominators, especially when dealing with roots and powers.

Consider the expression \(x^4\), which is under a square root sign as \(\sqrt{x^4}\). By the properties of exponents, specifically \((x^m)^n = x^{m\cdot n}\), we know this can be expressed as \((x^4)^{1/2}\), which simplifies to \(x^{4/2}\) or simply \(x^2\).

Applying similar logic to \(y^4\) under the square root, we find \(\sqrt{y^4} = y^{4/2} = y^2\). These principles allow us to efficiently solve and maneuver through equations. Remember, understanding these properties makes it easier to work with complex expressions and helps reduce the potential for mistakes.

Expression Simplification

The final step of simplifying the expression involves combining our previously simplified parts. Once we have broken down an expression and simplified its components, the next step is to evaluate and finalize it.

Taking the combined simplification \(4x^2y^2\), we place it back into the outer square root: \(\sqrt{4x^2y^2}\). By applying the same square root property as before, we break it down into manageable parts:

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

As a result, the simplified expression becomes \(2xy\).

This step shows how calculating square roots separately and then combining the results can transform a complex expression into a simpler form. Simplification not only makes expressions more understandable but also easier to work with in further mathematical operations.