Problem 77 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{7}{\sqrt{24 b^{3}}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{7\sqrt{6b}}{12b^2} \).

Step 1: Simplify the Radicand

First, factor the number inside the square root. We have \( \sqrt{24b^3} \). Note that \( 24 = 2^3 \times 3 \) and \( b^3 = b^2 \times b \). Thus, \( \sqrt{24b^3} = \sqrt{(2^3 \times 3) \times (b^2 \times b)} \).

Step 2: Separate and Simplify the Square Roots

Now, separate the square root into two parts: \( \sqrt{(2^3 \times 3)} \times \sqrt{(b^2 \times b)} = \sqrt{8} \times \sqrt{3} \times \sqrt{b^2} \times \sqrt{b} \). Simplify further as \( \sqrt{8} = 2\sqrt{2} \) and \( \sqrt{b^2} = b \). Thus, \( \sqrt{24b^3} = 2b\sqrt{6b} \).

Step 3: Rationalize the Denominator

We begin with \( \frac{7}{2b\sqrt{6b}} \). To rationalize, multiply the numerator and denominator by \( \sqrt{6b} \): \[ \frac{7 \times \sqrt{6b}}{2b \sqrt{6b} \times \sqrt{6b}} = \frac{7 \sqrt{6b}}{2b \times 6b} = \frac{7 \sqrt{6b}}{12b^2}. \]

Step 4: Final Simplified Expression

The rationalized expression is \( \frac{7\sqrt{6b}}{12b^2} \). All square roots have been removed from the denominator, achieving the goal of rationalization.

Key Concepts

Simplifying Radicals

When working with radicals, especially in the context of rationalizing denominators, simplifying radicals is a crucial first step. Radicals, often represented by the square root (\(sqrt{}\)) symbol, house expressions that aren't initially straightforward to combine or manipulate.
Simplifying them involves breaking down the number or expression inside the radical into its prime components.

• For example, with the square root of 24 (\(\sqrt{24}\)), we factor it into prime numbers. Since 24 is equal to \(2^3 \times 3\), it simplifies to \(\sqrt{(2^3 \times 3)}\).
• When variables are involved, as with \(b^3\), you split it similarly: \(b^3 = b^2 \times b\).

By breaking down expressions in this way, you can easily identify perfect squares and pairs that can come out of the radical, further simplifying your expression. This simplification paves the path to effectively rationalizing your denominator.

Square Roots

Square roots play an essential role in simplifying and manipulating mathematical expressions, especially when dealing with rational expressions. The square root of a number or expression is what, when multiplied by itself, gives the original number.
Understanding this concept is vital for rationalizing denominators.

• For instance, when you have \(\sqrt{b^2}\), it simplifies to \(b\) because \(b \times b = b^2\).
• This principle becomes handy when handling more complex expressions like \(\sqrt{24 b^3}\). Once simplified into its components—both numeric and variable—it becomes easier to manage and solve.

In more advanced operations, square roots allow for breaking apart expressions into manageable units that can be manipulated separately, making the entire process less daunting. Understanding square roots helps in systematically dismantling expressions inside radicals, thereby easing the rationalization process.

Rational Expressions

Rational expressions often consist of numerators and denominators that include square roots. Rationalizing such expressions is essential because mathematical norms prefer denominators to be free of roots. This makes expressions easier to work with both numerically and algebraically.
For an expression like \(\frac{7}{\sqrt{24 b^3}}\), and after simplifying radicals, the next step is rationalization.

• Once a square root is simplified, as shown in previous steps to \(2b\sqrt{6b}\), it's time to clear out the radicals from the denominator by multiplying both the numerator and the denominator by the conjugate or radical part that needs eliminating.
• By multiplying by \(\sqrt{6b}\) in this context, the denominator turns from \(2b\sqrt{6b}\) into \(12b^2\), removing the square root and simplifying the fraction.

Rationalizing denominators leads to a cleaner and often more straightforward form. This practice ensures that subsequent mathematical operations on the expression will be accurate and straightforward to handle, fitting the mathematical standard for presenting rational expressions.