Problem 86 - Algebra A Combined Function

Question

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. \(\frac{5}{\sqrt{27}}\)

Step-by-Step Solution

Answer

Multiply both by \(\sqrt{3}\).

Step 1: Identify the Denominator

The given expression is \(\frac{5}{\sqrt{27}}\). The denominator here is \(\sqrt{27}\). To rationalize the denominator, we need to remove the square root.

Step 2: Find the Smallest Number to Multiply

To eliminate the square root from the denominator, multiply both the numerator and the denominator by \(\sqrt{27}\). However, since \(27 = 3 \times 9 = 3 \times 3 \times 3 = 3^3\), \(\sqrt{27} = 3\sqrt{3}\). So we will multiply by \(\sqrt{3}\) which is \(\sqrt{27}\) divided by its integer factor \(3\).

Step 3: Multiply and Simplify

Multiply the expression \(\frac{5}{\sqrt{27}}\) by \(\frac{\sqrt{3}}{\sqrt{3}}\), turning it into \(\frac{5\sqrt{3}}{\sqrt{81}}\). Since \(\sqrt{81} = 9\), the expression becomes \(\frac{5\sqrt{3}}{9}\).

Step 4: Conclude

The smallest number to multiply both the numerator and the denominator by to remove the radical from the denominator is \(\sqrt{3}\). The resulting expression is fully rationalized.

Key Concepts

Square Roots

Square roots are fundamental in mathematics, especially when dealing with radicals in fractions. A square root of a number is a value which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 multiplied by 3 equals 9. In expressions like \( \frac{5}{\sqrt{27}} \), the denominator features a square root, which makes it an irrational expression. When simplifying such expressions, our goal is to transform the square root in the denominator into a rational number. This is done by a process called rationalization, which means finding a suitable number to multiply with both the numerator and the denominator to "remove" the square root.

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form. It's a crucial step when rationalizing the denominator of a fraction that involves radicals. For example, the square root of 27, namely \( \sqrt{27} \), can be simplified. Breaking it down to its prime factors gives us \( \sqrt{3 \times 3 \times 3} \), which is equivalent to \( 3\sqrt{3} \). This simplification shows us that the square root of 27 can be expressed as the product of 3 and the square root of 3, making it easier to handle when performing calculations. Simplifying the radical is necessary before further manipulating the expression, such as rationalizing the denominator. This helps ensure that calculations are accurate and completed with fewer steps.

Numerator and Denominator Multiplication

In the process of rationalizing an expression like \( \frac{5}{\sqrt{27}} \), multiplying both the numerator and the denominator by the same number is key. This keeps the expression equivalent to its original value while changing its form. For our example, we multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \), effectively "canceling out" the radical in the denominator.Here's how it works:

• Multiply the numerator: \( 5 \times \sqrt{3} = 5\sqrt{3} \)
• Multiply the denominator: \( \sqrt{27} \times \sqrt{3} = \sqrt{81} \)

This results in \( \frac{5\sqrt{3}}{9} \) since \( \sqrt{81} = 9 \). This method of multiplication ensures that the expression remains unchanged in value while achieving the goal of rationalizing the denominator. It's a straightforward approach that underscores the balance of mathematical operations in preserving equivalent expressions.