Problem 73 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt[3]{4 a^{6}}}{\sqrt[3]{2 a^{5} b}} $$

Step-by-Step Solution

Answer

\( \frac{ \sqrt[3]{16b^2} }{2b} \)

Step 1: Identify the Denominator

Identify that the denominator is \( \sqrt[3]{2a^{5}b} \). Our goal is to eliminate this cube root from the denominator by rationalizing it.

Step 2: Determine the Conjugate

To rationalize \( \sqrt[3]{2a^{5}b} \), recognize that its cube is \( (\sqrt[3]{2a^{5}b})^3 = 2a^{5}b \). In order to completely rationalize, multiply and divide the expression by \( (\sqrt[3]{2a^{5}b})^2 = \sqrt[3]{(2a^{5}b)^2} = \sqrt[3]{4a^{10}b^2} \).

Step 3: Multiply to Rationalize

Multiply the numerator and the denominator by \( \sqrt[3]{4a^{10}b^2} \):\[\frac{ \sqrt[3]{4a^{6}} \cdot \sqrt[3]{4a^{10}b^2} }{\sqrt[3]{2a^{5}b} \cdot \sqrt[3]{4a^{10}b^2} }\]This simplifies the denominator to \( \sqrt[3]{(2a^{5}b)^3} = 2a^{5}b \).

Step 4: Simplifying the Numerator

The expression in the numerator becomes:\[\sqrt[3]{4a^{6}} \cdot \sqrt[3]{4a^{10}b^2} = \sqrt[3]{16a^{16}b^2}\]This results in \( a^{5} \cdot \sqrt[3]{16b^2} \) because \( a^{16} = a^5 \cdot a \) simplifies to \( a^{5} \).

Step 5: Final Simplification

Substitute the simplified numerator and denominator:\[\frac{ a^{5} \cdot \sqrt[3]{16b^2} }{2a^{5}b}\]Cancel out \( a^{5} \) from the numerator and denominator:\[\frac{ \sqrt[3]{16b^2} }{2b}\]

Step 6: Conclusion

Therefore, the rationalized form of \( \frac{\sqrt[3]{4a^{6}}}{\sqrt[3]{2a^{5}b}} \) is \( \frac{ \sqrt[3]{16b^2} }{2b} \).

Key Concepts

Cube Roots

A cube root is a number that, when multiplied by itself three times, gives a particular product. In our exercise, we have cube roots like \( \sqrt[3]{4a^{6}} \) and \( \sqrt[3]{2a^{5}b} \). A cube root looks similar to a square root, but instead of a small 2 above the root sign, there is a 3. This shows that it’s about finding the number that needs to be multiplied

• three times (cubed) to achieve the value beneath the root.
• Think of it like solving a puzzle to find the original number that fits perfectly into a cube to reach the specified value.

Understanding cube roots helps us work through many mathematical problems, including rationalizing denominators like in our exercise.
To simplify a trickier cube root, first look for factors that can be cubed easily. Cube roots have applications not just in school math, but in real-life situations too, such as calculating volume for three-dimensional objects.

Simplifying Expressions

Simplifying expressions is like tidying up a room. We want to make our expressions as straightforward as they can be. When you have expressions with cube roots, simplifying means making them easier to handle or combining them into a single term.

• Consider our fraction from the exercise: \( \frac{ \sqrt[3]{4a^{6}} \cdot \sqrt[3]{4a^{10}b^2} }{\sqrt[3]{2a^{5}b} \cdot \sqrt[3]{4a^{10}b^2} } \).
• Simplifying helps remove the root from the denominator by turning the expression into a more elegant format that is easier to interpret and use in further calculations.

During the simplification process, especially with cube roots, break it into manageable pieces:

• Multiply like terms together.
• Simplify powers inside and outside of the root.

When we can cancel out the terms that exist in both the top and bottom of our fraction, it’s much neater and makes further work with the expression more straightforward.

Exponents and Powers

Exponents and powers are handy tools in math that show how many times a number, called the base, is multiplied by itself. In the exercise, variables like \( a^{6} \) and \( a^{5} \) have exponents, which simplify the way we write repeated multiplication. With exponents:

• \( a^{6} \) represents \( a \times a \times a \times a \times a \times a \).
• When simplifying expressions, use exponent rules, like \( a^{m} \times a^{n} = a^{m+n} \) and \( (a^{m})^{n} = a^{m \cdot n} \).

Exponents make handling larger values and mathematical expressions easier, especially for expressing and solving real-life problems. Understanding powers lets us simplify, rationalize, and manipulate expressions effectively, as we converted \( \sqrt[3]{16b^2} \) nicely back into the rationalized form in our solution.
Keeping a keen eye on these powers while working through any math problem helps to streamline the computation process efficiently.