Problem 53 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{\sqrt{18}}{\sqrt{27}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{6}}{3}\)

Step 1: Simplify the Numerator Radical

First, let's simplify the numerator, \(\sqrt{18}\). The prime factorization of 18 is \(2 \times 3^2\). Therefore, \(\sqrt{18} = \sqrt{2 \times 3^2} = \sqrt{2} \times \sqrt{3^2}\). This simplifies to \(3\sqrt{2}\).

Step 2: Simplify the Denominator Radical

Now, let's simplify the denominator, \(\sqrt{27}\). The prime factorization of 27 is \(3^3\). Therefore, \(\sqrt{27} = \sqrt{3^3} = \sqrt{3^2} \times \sqrt{3}\). This simplifies to \(3\sqrt{3}\).

Step 3: Simplify the Entire Fraction

We now simplify the fraction \(\frac{3\sqrt{2}}{3\sqrt{3}}\). Divide both the numerator and the denominator by 3, obtaining \(\frac{\sqrt{2}}{\sqrt{3}}\).

Step 4: Rationalize the Denominator

To remove the square root in the denominator, multiply the numerator and the denominator by \(\sqrt{3}\). Thus, \(\frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{2} \cdot \sqrt{3}}{3} = \frac{\sqrt{6}}{3}\).

Key Concepts

Prime Factorization

Prime factorization is a technique used to express a number as the product of its prime numbers. It is a fundamental tool for simplifying radicals and many other mathematical operations. When dealing with square roots, knowing the prime factorization of a number helps in breaking the number down into smaller, manageable parts. For example, the number 18 can be expressed as the prime factors of 2 and 3 squared, or simply, \(2 \times 3^2\).
By identifying these factors, we can simplify a square root like \(\sqrt{18}\) more easily. Since \(18 = 2 \times 3^2\), we have \(\sqrt{18} = \sqrt{2 \times 3^2}\). We can further break it to \(\sqrt{2} \times \sqrt{3^2}\) and since \(\sqrt{3^2} = 3\), it simplifies \(\sqrt{18}\) to \(3\sqrt{2}\).
Prime factorization allows one to handle large numbers and their roots by simplifying them to their basic building blocks, making the numbers more manageable and easily simplified.

Rationalizing the Denominator

Rationalizing the denominator is a method used to eliminate the square root or any radical from the denominator of a fraction. This makes the fraction easier to work with in its simplest form.
When we had the fraction \(\frac{\sqrt{2}}{\sqrt{3}}\), the denominator contained a radical. To rationalize it, we multiply both the numerator and the denominator by the radical present.

• Multiply top and bottom by \(\sqrt{3}\)
• This results in \(\frac{\sqrt{2} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}\)
• This simplifies to \(\frac{\sqrt{6}}{3}\)

Now, the denominator is rational, with no radicals remaining. Rationalizing is important as it standardizes expressions and facilitates further calculations by ensuring the denominator isn't a surd or a radical, ultimately making expressions clearer and more usable.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than one. Simplicity in fractions makes handling computations more straightforward and clear.
In our example, we simplified \(\frac{3\sqrt{2}}{3\sqrt{3}}\) by cancelling terms. Both the numerator and the denominator are divisible by 3, hence simplifying it to \(\frac{\sqrt{2}}{\sqrt{3}}\).

• Identify common factors in the numerator and the denominator.
• Divide both by the greatest common factor.

Once simplified, the expression becomes much easier to work with, whether it is involved in further calculations or evaluations.
By simplifying fractions, mathematical expressions become neater and direct, which is essential for clarity and precision in problem-solving.