Problem 88
Question
Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt{9}}{\sqrt[3]{3}}\)
Step-by-Step Solution
Verified Answer
The simplest radical form is \( \sqrt[3]{9} \).
1Step 1: Simplify the numerator
The numerator of the expression \( \sqrt{9} \) can be simplified. Since \( \sqrt{9} = 3 \), the numerator becomes \( 3 \).
2Step 2: Simplify the denominator
Now consider the denominator \( \sqrt[3]{3} \). This is already in its simplest form and cannot be simplified further.
3Step 3: Rationalize the denominator
To remove the radical from the denominator, we'll multiply both the numerator and the denominator by \( \sqrt[3]{9} \). This results in: \( \frac{3 \cdot \sqrt[3]{9}}{\sqrt[3]{3} \cdot \sqrt[3]{9}} = \frac{3 \sqrt[3]{9}}{3} \).
4Step 4: Simplify the expression
Cancel out the 3 in the numerator and the denominator: \( \sqrt[3]{9} \). This is already in simplest radical form.
Key Concepts
Numerator and DenominatorRationalizing the DenominatorCube Roots
Numerator and Denominator
When dealing with fractions, understanding the numerator and denominator is crucial. These are the two main components of any fraction. The numerator is the top part of the fraction and it indicates the number of parts you have. The denominator is the bottom part and it tells you how many parts make up a whole.
In the exercise, we started with \( \frac{\sqrt{9}}{\sqrt[3]{3}} \). The numerator here was \( \sqrt{9} \), which signifies the square root of 9. When we simplify \( \sqrt{9} \), it becomes 3. Thus, the numerator is easily transformed for clearer results.
Understanding how to work with numerators and denominators allows you to manipulate the expression appropriately and reach the simplest form possible. This step prepares us to work further with rationalizing the denominator.
In the exercise, we started with \( \frac{\sqrt{9}}{\sqrt[3]{3}} \). The numerator here was \( \sqrt{9} \), which signifies the square root of 9. When we simplify \( \sqrt{9} \), it becomes 3. Thus, the numerator is easily transformed for clearer results.
Understanding how to work with numerators and denominators allows you to manipulate the expression appropriately and reach the simplest form possible. This step prepares us to work further with rationalizing the denominator.
Rationalizing the Denominator
Rationalizing the denominator is a process we use when we want to eliminate radicals from the bottom of a fraction. This is done to make the fraction cleaner and easier to work with.
In our example, we had the expression \( \frac{3}{\sqrt[3]{3}} \). The denominator had a cube root, \( \sqrt[3]{3} \), which we wanted to remove. To achieve this, we multiplied both the numerator and the denominator by a number that could cancel the cube root in the denominator.
Here's how it works:
It makes expressions easier to read and work with in both computations and applications.
In our example, we had the expression \( \frac{3}{\sqrt[3]{3}} \). The denominator had a cube root, \( \sqrt[3]{3} \), which we wanted to remove. To achieve this, we multiplied both the numerator and the denominator by a number that could cancel the cube root in the denominator.
Here's how it works:
- Multiply both parts of the fraction by the cube root of 9, or \( \sqrt[3]{9} \).
- This resulted in \( \frac{3 \cdot \sqrt[3]{9}}{\sqrt[3]{3} \cdot \sqrt[3]{9}} \), which simplifies further.
It makes expressions easier to read and work with in both computations and applications.
Cube Roots
Cube roots offer a unique twist to simplification, especially when compared to square roots. A cube root, represented by \( \sqrt[3]{x} \), asks what number squared is equal to \( x \).
In our expression, the cube root \( \sqrt[3]{3} \) is the central element. While you cannot simplify \( \sqrt[3]{3} \) further directly, understanding its properties allows you to use it effectively in expressions.
When we multiplied by \( \sqrt[3]{9} \) to rationalize the expression, we aimed to create perfect cubes that allow for further simplification. When combining cube roots:
Cube roots, though trickier than basic square roots, can be navigated with these principles to keep expressions elegant and simplified.
In our expression, the cube root \( \sqrt[3]{3} \) is the central element. While you cannot simplify \( \sqrt[3]{3} \) further directly, understanding its properties allows you to use it effectively in expressions.
When we multiplied by \( \sqrt[3]{9} \) to rationalize the expression, we aimed to create perfect cubes that allow for further simplification. When combining cube roots:
- \( \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab} \)
Cube roots, though trickier than basic square roots, can be navigated with these principles to keep expressions elegant and simplified.
Other exercises in this chapter
Problem 86
Explain how to simplify \(\left(2^{-1} \cdot 3^{-2}\right)^{-1}\) and also how simplify \(\left(2^{-1}+3^{-2}\right)^{-1}\).
View solution Problem 87
Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{8}}{\sqrt[4]{4}}\)
View solution Problem 88
Use a calculator to simplify each of the following numerical expressions. Express your answers to the nearest hundredth. (a) \(\left(2^{-3}+3^{-3}\right)^{-2}\)
View solution Problem 89
Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[4]{27}}{\sqrt{3}}\)
View solution