Problem 87 - Algebra for College Students

Question

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{8}}{\sqrt[4]{4}}\)

Step-by-Step Solution

Answer

\(\sqrt{2}\)

Step 1: Simplify the numerator

We begin by simplifying the expression in the numerator \(\sqrt[3]{8}\). Note that \(8\) can be expressed as \(2^3\), therefore \(\sqrt[3]{8} = \sqrt[3]{2^3} = 2\).

Step 2: Simplify the denominator

Next, simplify the denominator \(\sqrt[4]{4}\). The number \(4\) can be written as \(2^2\), so the expression becomes \(\sqrt[4]{2^2}\). Since the fourth root of \(2^2\) does not simplify to an integer, we write it as \((2^2)^{1/4} = 2^{1/2} = \sqrt{2}\).

Step 3: Divide the expressions

Now, divide the simplified numerator by the simplified denominator: \(\frac{2}{\sqrt{2}}\).

Step 4: Rationalize the denominator

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{2}\):\[\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2}\]

Step 5: Simplify the expression

Finally, simplify \(\frac{2\sqrt{2}}{2}\) to obtain \(\sqrt{2}\). The twos in both the numerator and the denominator cancel each other out.

Key Concepts

Understanding Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives you the original number. For instance, when you have the cube root of 8, you're searching for a number which when cubed (that is, multiplied by itself three times) results in 8. Let's look at 8. We know that 8 is equal to \(2^3\). Therefore, \(\sqrt[3]{8}\) simplifies to 2 because 2 multiplied by itself, three times, equals 8:

• \(2 \times 2 \times 2 = 8\)

Recognizing this multiplication pattern makes it easier to grasp cube roots, especially when working with radicals. Always try to express the number inside the radical as a power of some smaller number to make your calculations simpler.

Delving Into Fourth Roots

The fourth root of a number looks for a value that, when multiplied by itself four times, equals the original number. In our exercise, we work with the number 4. We want to figure out \(\sqrt[4]{4}\). If we break down 4, we get \(4 = 2^2\). Even though when we think about multiplying \(2\) four times (\(2 \times 2 \times 2 \times 2\)), it's not the same as our \(2^2\), hence:\[\sqrt[4]{2^2} = (2^2)^{1/4} = 2^{1/2} = \sqrt{2}\]This means that \(\sqrt{2}\) is the simplest radical form of the fourth root of 4. Fourth roots might initially seem tricky, but the key is to express your number as a power. This simplification makes it manageable.

Rationalizing the Denominator

In mathematics, simplifying expressions in radical form is often preferred. However, if you have a radical in the denominator, we aim to "rationalize" it. This means we want to remove the radical from the denominator.In our example:

• You have \(\frac{2}{\sqrt{2}}\).

To rationalize the denominator, multiply both the numerator and denominator by \(\sqrt{2}\):\[\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2}\]This step eliminates the radical in the denominator, leaving you with the expression \(\sqrt{2}\). The rule is straightforward: multiply by a form of 1 (like \(\frac{\sqrt{2}}{\sqrt{2}}\) in this case) to simplify. This process improves the aesthetics of your answers, making them easier to work with and understand.