Problem 73 - Elementary and 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

The rationalized expression is \( \frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}} \).

Step 1: Simplify the Radicals

We have the expression \( \frac{\sqrt[3]{4 a^{6}}}{\sqrt[3]{2 a^{5} b}} \). First, simplify each cube root. \( \sqrt[3]{4 a^6} = \sqrt[3]{4} \cdot \sqrt[3]{a^6} = \sqrt[3]{4} \cdot a^2 \) since \(a^6 = (a^2)^3\). Similarly, \( \sqrt[3]{2 a^5 b} = \sqrt[3]{2} \cdot \sqrt[3]{a^5} \cdot \sqrt[3]{b} = \sqrt[3]{2} \cdot a^{\frac{5}{3}} \cdot b^{\frac{1}{3}} \).

Step 2: Combine and Reduce the Fraction

Now substitute the simplified forms back into the original expression: \( \frac{\sqrt[3]{4} \cdot a^2}{\sqrt[3]{2} \cdot a^{\frac{5}{3}} \cdot b^{\frac{1}{3}}} \). Divide the \(a\) terms: \( \frac{a^2}{a^{\frac{5}{3}}} = a^{\frac{6}{3} - \frac{5}{3}} = a^{\frac{1}{3}} \). Thus, the expression simplifies to \( \frac{\sqrt[3]{4} \cdot a^{\frac{1}{3}}}{\sqrt[3]{2} \cdot b^{\frac{1}{3}}} \).

Step 3: Rationalize the Denominator

To rationalize the denominator, we need to multiply both the numerator and the denominator by \( \sqrt[3]{4} \cdot \sqrt[3]{b^{\frac{2}{3}}} \) so that the cube root in the denominator becomes a whole number. This results in: \[ \frac{\sqrt[3]{4} \cdot a^{\frac{1}{3}} \cdot \sqrt[3]{2} \cdot \sqrt[3]{b^{\frac{2}{3}}}}{\left(\sqrt[3]{2} \cdot \sqrt[3]{b^{\frac{1}{3}}}\right) \cdot \sqrt[3]{2} \cdot \sqrt[3]{b^{\frac{2}{3}}}} \] Upon simplification, the denominator becomes \(2b\), leaving us with \[ \frac{\sqrt[3]{8} \cdot a^{\frac{1}{3}} \cdot b^{\frac{2}{3}}}{2b} \].

Step 4: Simplify the Result

Since \( \sqrt[3]{8} = 2 \), the expression further simplifies to: \[ \frac{2 \cdot a^{\frac{1}{3}} \cdot b^{\frac{2}{3}}}{2b} \] \[ = \frac{a^{\frac{1}{3}} \cdot b^{\frac{2}{3}}}{b} \] \[ = a^{\frac{1}{3}} \cdot b^{-\frac{1}{3}} \] which equals \( \frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}} \).

Step 5: Final Simplified Expression

Thus, the completely rationalized expression is: \( \frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}} \).

Key Concepts

Simplifying Radicals

When we simplify radicals, we aim to express them in the simplest form possible by removing any perfect powers under the root. This involves breaking down a number or expression inside a radical into its factors to see if any part of it is a perfect power of the root. For example, in cube roots, we seek to find perfect cubes within the number.

In the given exercise, we initially had the expression \( \sqrt[3]{4 a^6} \text{ and } \sqrt[3]{2 a^5 b} \). Simplification involves assessing whether these components can be written in an easier form. For \( \sqrt[3]{a^6} \), we could rewrite this as \((a^2)^3\) and thus it simplifies to \(a^2\), since the cube root and cube cancel each other out.

Similarly, for \( \sqrt[3]{a^5} \), by expressing it as \(a^{\frac{5}{3}}\), we maintain the simpler fractional form of exponents, which makes it manageable during further operations. This process of simplification helps in smooth progression toward solving or manipulating the expression further.

Cube Roots

Cube roots are the opposite of cube powers. When calculating the cube root, you're essentially asking, "what number, when multiplied three times, gives the original number?" It’s denoted by \( \sqrt[3]{...} \). This operation is crucial when working to simplify expressions or in rationalizing denominators involving cube roots.

For instance, consider \( \sqrt[3]{8} \). This is asking: "Which number multiplied by itself three times equals 8?" The answer, from basic understanding, is 2.

• \(2 \times 2 \times 2 = 8\)

Similarly, when working through expressions with cube roots like \( \sqrt[3]{4} \), since 4 is not a perfect cube, it remains under the radical until further simplified by additional expressions.

Understanding cube roots helps when trying to rationalize denominators, like in the exercise where expressions were multiplied to eliminate the radicals from the denominator, simplifying to more manageable terms.

Exponents

Exponents represent repeated multiplication. They're crucial in breaking down and simplifying expressions, particularly involving radicals. An exponent, displayed as a superscript number (^{n}), indicates how many times a number (base) is used in a multiplication. This connects deeply with radicals too, as roots can be expressed as fractional exponents.

In the context of the exercise, note the conversion of radical expressions to exponents. For \( a^6 \), using the property \( a^{m/n} = \sqrt[n]{a^m} \), this converts to \((a^2)^3\). Additionally, fractional exponents like \( a^{5/3} \) simplify operations, especially when dividing or subtracting in expression manipulations.

• e.g., \( a^2 / a^{5/3} = a^{2 - 5/3} \).

By having a grasp on exponents, including managing fractional exponents, tackling complex expressions becomes attainable and straightforward. This foundational knowledge helps simplify the overall computation, making the rationalization of denominators effective and efficient.