Problem 23 - College Algebra

Question

Use the quotient rule to simplify the expressions in Exercises \(17-26 .\) Assume that \(x>0\) $$\frac{\sqrt{150 x^{4}}}{\sqrt{3 x}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{\sqrt{150 x^{4}}}{\sqrt{3 x}}\) is \(5x\sqrt{2x}\)

Step 1: Apply the Quotient Rule

Start by applying the quotient rule to the given expression \(\frac{\sqrt{150 x^{4}}}{\sqrt{3x}},\) to give a new expression \(\sqrt{\frac{150x^{4}}{3x}}\)

Step 2: Simplify the Fraction

Next, simplify the fraction under the square root by simplifying \(150x^{4}\) divided by \(3x\) to get \(50x^{3}\). So it becomes: \(\sqrt{50x^{3}}.\)

Step 3: Break Down the Square Root

Break down the expression under the square root into its components. This can be done by recognizing that \(50\) is \(25*2\) and \(x^{3}\) is \(x^{2}*x\). Expressing our equation in this way, it becomes \(\sqrt{25*2*x^{2}*x}.\)

Step 4: Simplify the Square Root

Now, simplify the square root by taking the square root of any perfect squares. \(25\) is a perfect square, and so is \(x^{2}\), thus our expression becomes \(5x\sqrt{2x}.\)

Key Concepts

Radical Expressions

Radical expressions are commonly used in algebra to solve equations or simplify expressions. They involve numbers or variables under a radical sign, often a square root. Here, we're working with the expression \( \sqrt{150x^{4}} \) in the numerator and \( \sqrt{3x} \) in the denominator.

Radicals can be tricky, but the key is to understand how to manipulate them. You can use the property \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \) to combine or separate radicals. In the exercise, combining the radicals into a single square root made it easier to simplify the expression.

When dealing with radicals:

• Look for perfect squares you can simplify.
• Combine or separate terms under the radical as needed.
• Remember that simplifying radicals sometimes involves factoring the number or expression under the square root.

Simplification

Simplification is about making expressions easier to work with by breaking them down into their simplest form. In this exercise, using the quotient rule helped simplify the equation by turning a division of two radicals into a single radical expression. This is the first step towards simplification.

Once you've re-written the expression, look for opportunities to simplify terms inside. Divide the terms inside or factor them as shown in the expression \( \frac{150x^{4}}{3x} = 50x^{3} \). This type of simplification makes further reduction of the equation into its simplest terms much more manageable.

Key steps to consider:

• Simplify inside the radical by factoring out common terms.
• Look for perfect squares, which are easier to work with and often present in algebraic simplification tasks.
• Break down complex terms into simpler components.

Algebraic Fractions

An algebraic fraction is an expression that includes polynomials in the numerator, the denominator, or both. These fractions operate similarly to numeric fractions, meaning you can simplify by dividing out common factors.

In this exercise, the algebraic fraction is \( \frac{150x^{4}}{3x} \). Simplifying it means finding the greatest common factor (GCF) of the terms in the numerator and the denominator and canceling it out. This simplification leads to taking just a part of \( x \) in the denominator to make the expression \( 50x^{3} \).

Remember when dealing with algebraic fractions:

• Always factor out the greatest common factor to simplify fractions.
• Reduce the coefficients as you would in regular fractions: divide both the numerator and the denominator by their GCF.
• If variables are involved, reduce by canceling out the highest power common to both the numerator and denominator.