Problem 25 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{72 x^{2} y^{4}}$$

Step-by-Step Solution

Answer

The simplified expression is \(6xy^2\sqrt{2}\).

Step 1. Factor the Expression Under the Radical

First, we look to factor the number inside the square root. Begin with 72: note that it can be factored as \(72 = 2^3 \times 3^2\). Next, for the variables, we have \(x^2\) and \(y^4\). Thus, the entire expression under the square root can be rewritten as \(\sqrt{2^3 \times 3^2 \times x^2 \times y^4}\).

Step 2. Apply the Square Root to Each Factor

The square root of a multiplication can be expressed as the multiplication of square roots. Therefore, we rewrite the expression as \(\sqrt{2^3} \cdot \sqrt{3^2} \cdot \sqrt{x^2} \cdot \sqrt{y^4}\).

Step 3. Simplify Each Square Root

Now, take the square root of each individual component: \(\sqrt{2^3} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2}\), then \(\sqrt{3^2} = 3\), \(\sqrt{x^2} = x\), and \(\sqrt{y^4} = y^2\). Thus, the expression becomes \(2 \sqrt{2} \cdot 3 \cdot x \cdot y^2\).

Step 4. Multiply the Simplified Terms

Multiply the constants and the remaining terms outside of the square root: \(2 \cdot 3 \cdot x \cdot y^2 = 6xy^2\). Thus, the simplified expression is \(6xy^2\sqrt{2}\).

Key Concepts

Simplifying Radicals

When we simplify a radical expression, we aim to express it in a more concise understanding without a fractional or radical component under the radical sign. This involves reducing the radicand, or the number under the square root, to its prime factors and simplifying any perfect squares.

• Identify and factorize the components under the radical.
• Split the expression into separate square roots if possible.
• Extract and simplify any perfect square factors.

By understanding these steps, you're able to reduce complex expressions. For instance, if you begin with a number like 72, you will factor it into 2 cubed and 3 squared. The next step would be to simplify these factors by pulling out perfect squares, like 4 from 2 cubed, making it easier to handle. Through practice, simplifying radicals becomes a straightforward process exploding the expression to its cleanest form.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It's represented by the radical symbol \( \sqrt{} \) . Being able to find square roots is crucial for simplifying radical expressions. Consider the square root of perfect square numbers such as 4 and 9, which are 2 and 3, respectively.

• Recall that the square root essentially undoes a square operation.
• Focus on always simplifying square roots where possible.
• Simplifying a square root helps in making expressions less cumbersome.

For each factor inside a square root, simplify components where you can completely take the root, like \( \sqrt{4} = 2 \). If the number or expression isn’t a perfect square, sometimes leaving part of the expression under the root is necessary to retain accuracy, as in the case of \( \sqrt{2} \).

Factoring Under Radicals

Factoring under a radical involves breaking down the expression to its core components. We aim to expose perfect squares which can be easily simplified when square roots are applied. For example, the radical expression \( \sqrt{72 x^2 y^4} \) involves breaking 72 into \( 2^3 \times\ 3^2 \), which highlights the perfect squares: \( 4 \times\ 9 \).

• Begin with prime factorization of the number inside the radical.
• Factor variable expressions like \( x^2 \) and \( y^4 \) to observe which can be simplified.
• Identify perfect squares that can be simplified under the radical.

Breaking an expression down allows us to see components that can be pulled out of the radical. For instance, \( y^4 \) is already a perfect square as \( (y^2)^2 \), making it directly simplifiable to \( y^2 \). Factoring exposes these structure points, revealing which portions to simplify for an accurate, concise expression.