Problem 69 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt{48 x^{2}}}{\sqrt{8 x^{2} y}} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{\sqrt{6y}}{y} \).

Step 1: Simplify the Numerator and Denominator

The given expression is \( \frac{\sqrt{48x^2}}{\sqrt{8x^2y}} \). First, simplify the numerator and the denominator separately. The numerator \( \sqrt{48x^2} \) can be simplified to \( \sqrt{16 \times 3 \times x^2} = \sqrt{16} \times \sqrt{3} \times \sqrt{x^2} = 4x\sqrt{3} \). The denominator \( \sqrt{8x^2y} \) can be simplified to \( \sqrt{4 \times 2 \times x^2 \times y} = \sqrt{4} \times \sqrt{2} \times \sqrt{x^2} \times \sqrt{y} = 2x\sqrt{2y} \). This gives us the expression \( \frac{4x\sqrt{3}}{2x\sqrt{2y}} \).

Step 2: Simplify the Fraction

Now that we have \( \frac{4x\sqrt{3}}{2x\sqrt{2y}} \), notice that we can simplify the fraction by canceling \( x \) in the numerator and the denominator, and also simplifying the numerical fraction \( \frac{4}{2} \). Therefore, the expression becomes \( \frac{2\sqrt{3}}{\sqrt{2y}} \).

Step 3: Rationalize the Denominator

To rationalize the denominator \( \sqrt{2y} \), multiply both the numerator and the denominator by \( \sqrt{2y} \). This results in: \[ \frac{2\sqrt{3}}{\sqrt{2y}} \times \frac{\sqrt{2y}}{\sqrt{2y}} = \frac{2\sqrt{6y}}{2y} \].

Step 4: Simplify the New Fraction

Simplify the fraction \( \frac{2\sqrt{6y}}{2y} \) by canceling the common factor of 2 in the numerator and the denominator. This results in \( \frac{\sqrt{6y}}{y} \). Thus, the rationalized expression is \( \frac{\sqrt{6y}}{y} \).

Key Concepts

Simplifying Radicals

Radicals are expressions that include a root, such as the square root. Simplifying radicals is the process of manipulating these expressions to their most basic form, which can help in performing arithmetic operations more easily. The given expression is \( \frac{\sqrt{48x^2}}{\sqrt{8x^2y}} \). This expression includes two separate square roots:

• The numerator: \( \sqrt{48x^2} \)
• The denominator: \( \sqrt{8x^2y} \)

For the numerator \( \sqrt{48x^2} \), we simplify it by breaking it down into \( \sqrt{16 \times 3 \times x^2} \). This is the product of perfect squares and other terms, resulting in \( 4x\sqrt{3} \) since \( 16 \) and \( x^2 \) have perfect square roots of \( 4 \) and \( x \) respectively.

Similarly, the denominator \( \sqrt{8x^2y} \) simplifies to \( \sqrt{4 \times 2 \times x^2 \times y} = 2x\sqrt{2y} \), using the same method of identifying and taking out perfect square roots. This simplification allows us to work more straightforwardly with the expression, making further arithmetic operations easier to manage.

Fraction Simplification

Once radicals in the numerator and denominator are simplified, the next step is fraction simplification. Simplifying fractions entails reducing the expression by cancelling common factors from the numerator and the denominator.In our expression \( \frac{4x\sqrt{3}}{2x\sqrt{2y}} \), the following common factors can be identified:

• The variable \( x \) in both the numerator and the denominator
• The common numerical factor of \( 2 \)

By removing these common factors, the expression simplifies to \( \frac{2\sqrt{3}}{\sqrt{2y}} \). This step is crucial because it further reduces the expression, making it much simpler and also preparing it for the next procedure of rationalization of the denominator.

Mathematical Expressions

In mathematics, expressions are composed of numbers, variables, and operations. The expression we have been working with is a fraction involving radicals. When handling such expressions, the goal is often to make them simpler to understand or easier to use. This can involve arithmetic operations, factorization, and rationalization.The last step put into practice in our exercise is rationalizing the denominator. The expression \( \frac{2\sqrt{3}}{\sqrt{2y}} \) has a radical in the denominator. To rationalize it, we multiply both the numerator and the denominator by \( \sqrt{2y} \). This manipulation is key because it removes the root from the denominator. The new form of the expression is \( \frac{2\sqrt{6y}}{2y} \).

Lastly, simplify the final fraction \( \frac{2\sqrt{6y}}{2y} \) by canceling the \( 2 \) in both the numerator and the denominator, resulting in the rationalized form \( \frac{\sqrt{6y}}{y} \). This series of operations makes the expression more manageable and aligned with mathematical conventions of expression simplification.