Problem 45 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{\frac{2}{3 x}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{\sqrt{6x}}{3x} \).

Step 1: Identify the Expression

The given expression to simplify is \( \sqrt{ \frac{2}{3x} } \). This is a square root of a fraction.

Step 2: Apply the Property of Radicals

Apply the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Therefore, we can rewrite the expression as \( \frac{\sqrt{2}}{\sqrt{3x}} \).

Step 3: Simplify the Denominator

Since \( 3x \) is a product inside the square root, split this using the product rule, yielding \( \frac{\sqrt{2}}{\sqrt{3} \cdot \sqrt{x}} \).

Step 4: Rationalize the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{3} \sqrt{x} \). This gives \( \frac{\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{x}}{\sqrt{3} \cdot \sqrt{x} \cdot \sqrt{3} \cdot \sqrt{x}} = \frac{\sqrt{6x}}{3x} \).

Step 5: Simplified Expression

The final simplified form of \( \sqrt{ \frac{2}{3x} } \) is \( \frac{\sqrt{6x}}{3x} \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number. Imagine a number line. The square root of a number finds a point on this line which, when squared, returns to the initial value.

• For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9.
• Square roots have both positive and negative values; however, we typically use the positive value, known as the principal square root.

When dealing with square roots of algebraic expressions, we can use properties of square roots, such as: \[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]This property allows us to break down complex fractions into simpler parts, enabling easier manipulation and computation in mathematical expressions.

Rationalizing Denominator

Rationalizing the denominator is a technique used to eliminate radicals from the bottom of a fraction. This makes expressions cleaner and easier to evaluate or further simplify.Imagine needing to calculate with a number having a radical in the denominator. It can become cumbersome and less intuitive. To rationalize:

• Multiply both the numerator and the denominator of the fraction by a suitable term that will eliminate the radical in the denominator.

For example, in the expression \( \frac{\sqrt{2}}{\sqrt{3x}} \), multiplying both parts by \( \sqrt{3x} \) helps clear out the radicals in the denominator:\[\frac{\sqrt{2} \cdot \sqrt{3x}}{\sqrt{3x} \cdot \sqrt{3x}} = \frac{\sqrt{6x}}{3x}\]Now the denominator is rationalized since it no longer contains a square root.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the fraction to its simplest form, making calculations more manageable and results more interpretable. This often requires factoring, distribution, or cancellation of common terms.To simplify an algebraic fraction:

• Identify and cancel any common factors present in both the numerator and the denominator.
• Look out for opportunities to simplify using algebraic identities or rules, such as distributing or combining like terms.

For example, after rationalizing \( \sqrt{\frac{2}{3x}} \) to \( \frac{\sqrt{6x}}{3x} \), it's important to check if further simplification is possible. In this case, since \( 6x \) and \( 3x \) share no additional common factors, we have reached the simplest form. Simplifying not only clarifies expressions but also ensures preciseness in solving mathematical problems.