Problem 41 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{\frac{27}{16}}\)

Step-by-Step Solution

Answer

\(\frac{3\sqrt{3}}{4}\)

Step 1: Simplify the Fraction Under the Radical

First, we identify that the expression inside the square root is a fraction: \(\frac{27}{16}\). We write this as \(\sqrt{27} \div \sqrt{16}\), which allows us to simplify each part separately.

Step 2: Simplify the Radicals

Calculate \(\sqrt{27}\) and \(\sqrt{16}\) separately. \(\sqrt{16}=4\) because 16 is a perfect square (\(4^2 = 16\)). For \(\sqrt{27}\), rewrite 27 as \(3^3 = 3 \times 9\), thus, \(\sqrt{27} = \sqrt{3 \times 3 \times 3} = 3\sqrt{3}\) because \(\sqrt{9} = 3\).

Step 3: Divide the Simplified Radical by the Integer

Now, divide the expression \(3\sqrt{3}\) (which is the simplified numerator) by \(4\) (the simplified denominator): \(\frac{3\sqrt{3}}{4}\).

Step 4: Combine to Express in Simplest Radical Form

The simplest radical form of the original expression \(\sqrt{\frac{27}{16}}\) is therefore \(\frac{3\sqrt{3}}{4}\) as we cannot simplify \(\sqrt{3}\) any further.

Key Concepts

Simplifying Fractions Under Radicals

Simplifying fractions under radicals involves breaking down a complex square root that contains a fraction. The process begins with separating the square root of a fraction into the square roots of its numerator and denominator. For example, if you have \( \sqrt{\frac{27}{16}} \), you can rewrite this as \( \frac{\sqrt{27}}{\sqrt{16}} \).
This method simplifies the process because you are dealing with smaller parts rather than the whole fraction at once.

• The square root of the numerator (in this case \(27\)) is handled separately from the denominator (\(16\)).
• Each radical is simplified on its own, which will lead to a simpler form for the overall expression.

By tackling each part individually, it becomes easier to manage and work with larger fractions under a root.

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. Recognizing perfect squares is critical in simplifying radicals because it allows us to simplify parts of the expression into integers. In our example \(\sqrt{\frac{27}{16}}\), the denominator \(16\) is a perfect square.

• Perfect squares, like \(16\) in the denominator, simplify to integers. Here, \(\sqrt{16} = 4\) because \(4^2 = 16\).
• These simplifications help reduce the complexity of the expression, making it less daunting.

Understanding perfect squares can significantly simplify your calculations, helping transform the radical expression into something more manageable.

Expressing in Simplest Radical Form

Expressing an expression in its simplest radical form means breaking it down to where it can't be simplified further. This means ensuring that under the radical, there are no perfect square factors left. In our example, once the radicals of the fraction \(\sqrt{\frac{27}{16}}\) have been simplified, the result is \(\frac{3\sqrt{3}}{4}\).

• Simplify each square root where possible, leaving no perfect square factors under the radical.
• Combine the simplified numerators and denominators to express the overall fraction.
• Ensure that there are no radicals left in the denominator for the expression to be considered simplest.

This practice not only offers a cleaner form but also one that's easier to work with in future calculations, should more operations be required.