Problem 24 - Technical Mathematics with Calculus

Question

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$3 \sqrt{50 x^{5}}$$

Step-by-Step Solution

Answer

\( 15x^{2} \sqrt{2x} \)

Step 1: Simplify the Radical

Begin by factoring the number inside the square root to find perfect squares. The number 50 can be factored into 25 and 2, where 25 is a perfect square. So, write \( 3 \sqrt{50 x^{5}} \) as \( 3 \sqrt{25 \times 2 x^{5}} \).

Step 2: Separate the Perfect Square

Separate the perfect square (25) and the variable with a perfect square exponent (\(x^{4}\)) from the rest. Rewrite the expression as \( 3 \sqrt{25} \sqrt{2} \sqrt{x^{4}} \sqrt{x} \) which becomes \( 3 \cdot 5 \cdot \sqrt{2} \cdot x^{2} \sqrt{x} \).

Step 3: Simplify the Expression

Multiply the numbers and simplify the variables outside the radical. The expression \( 3 \cdot 5 \cdot \sqrt{2} \cdot x^{2} \sqrt{x} \) simplifies to \( 15x^{2} \sqrt{2x} \).

Key Concepts

Perfect Squares

Understanding perfect squares is crucial when simplifying radicals, as it allows us to transform the radical into a simpler form. A perfect square is the product of any integer multiplied by itself. For instance, the number 4 is a perfect square because it is the result of multiplying 2 by 2. Similarly, 25 is a perfect square because it equals 5 multiplied by 5.

To simplify the radical expression, we begin by identifying perfect squares in the radicand—that's the number inside the radical. In our example, we have identified that 50 has a perfect square factor of 25. Recognizing perfect squares enables us to rewrite the radical in a way that separates the perfect square from non-perfect square factors. This process simplifies the task of root extraction, which is a key step towards simplifying the entire radical expression.

Radical Expressions

A radical expression involves roots, such as square roots and cube roots of numbers. Simplifying radical expressions is about expressing them in the most condensed and straightforward form without changing the value of the original expression. The technique often requires identifying and extracting perfect square factors from under the radical.

In the provided exercise, the radical expression is simplified by factoring out the perfect square (25) and separating powers of the variable (x) with perfect square exponents from those without. For instance, in the term \(x^5\), we separated \(x^4\), which is a perfect square, from \(x\) that isn’t. By doing that and using extraction of roots (\(\sqrt{25} = 5\) and \(\sqrt{x^4} = x^2\)), we can move from the radical to a simpler multiplication expression involving no radicals or simpler radicals.

Exponent Rules

Exponent rules are vital for simplifying expressions that include variables raised to a power. One such rule is that when a product of powers with the same base is under a radical, we can simplify it by taking the root of each power separately.

In the context of our exercise, this rule was applied to \(x^5\), which is written as \(x^4 \cdot x\). We treat \(x^4\) as a perfect square because any even power of x can be rewritten as a square of some expression, in this case, \(x^2\). After extracting the square root of the perfect square, the next step is to multiply the resulting expression outside the radical, simplifying the operation into basic multiplication. Following these exponent rules is essential in breaking down complex radical expressions into more manageable pieces.