Problem 75 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{23}{\sqrt{50 p^{5}}} $$

Step-by-Step Solution

Answer

\( \frac{23\sqrt{2p}}{10p^3} \) is the rationalized expression.

Step 1: Identify the Rationalization Target

In this exercise, we need to rationalize the denominator, which is \( \sqrt{50p^5} \). Rationalization involves removing the square root from the denominator.

Step 2: Simplify the Square Root in the Denominator

First, simplify \( \sqrt{50p^5} \). Break down the expression: \( 50 = 25 \times 2 \) and \( p^5 = p^4 \times p \). Thus, \( \sqrt{50p^5} = \sqrt{25 \times 2 \times p^4 \times p} = \sqrt{25} \times \sqrt{p^4} \times \sqrt{2p} \). Since \( \sqrt{25} = 5 \) and \( \sqrt{p^4} = p^2 \), this becomes \( 5p^2\sqrt{2p} \).

Step 3: Multiply to Rationalize

To rationalize, multiply both numerator and denominator by \( \sqrt{2p} \). This gives us: \[ \frac{23}{5p^2\sqrt{2p}} \times \frac{\sqrt{2p}}{\sqrt{2p}} = \frac{23\sqrt{2p}}{5p^2(\sqrt{2p})^2} \].

Step 4: Simplify the Expression

Calculate \((\sqrt{2p})^2 = 2p\). The denominator becomes \( 5p^2 \times 2p = 10p^3 \). The expression simplifies to \( \frac{23\sqrt{2p}}{10p^3} \).

Step 5: Verify Rationalization

The denominator is now \( 10p^3 \), which has no square roots, completing the rationalization process. Ensure no further simplification is possible in terms of the fractions or radicals.

Key Concepts

Square Roots

Square roots often pop up in algebra, especially in the context of rationalizing denominators. To grasp the idea of rationalizing, it's essential to understand square roots first. A square root of a number is another number that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because 5 times 5 equals 25.Square roots become more complex when variables are involved. Consider \( \sqrt{p^5} \). This can be broken down as \( \sqrt{p^4} \times \sqrt{p} \).

• \( \sqrt{p^4} \) simplifies to \( p^2 \), because \( p^2 \times p^2 = p^4 \).
• The square root \( \sqrt{p} \) stays as is because \( p \) does not have an even power that can be split equally.

By simplifying square roots, you can often make the process of dealing with complex rational expressions much simpler.

Simplifying Radical Expressions

Simplifying radical expressions involves breaking them down into simpler components. The ultimate goal with rationalization is to remove radicals from the denominators by making them simpler.To simplify the denominator \( \sqrt{50p^5} \), we first separate it into known radicals:

• Recognize 50 as \( 25 \times 2 \), so \( \sqrt{50} \) becomes \( \sqrt{25} \times \sqrt{2} \).
• Since \( \sqrt{25} = 5 \), this simplifies to \( 5 \times \sqrt{2} \).
• For \( \sqrt{p^5} \), express it as \( \sqrt{p^4} \times \sqrt{p} \) which simplifies to \( p^2 \times \sqrt{p} \).

The expression turns into \( 5p^2\sqrt{2p} \). This step is crucial because it prepares the expression for the rationalization process by eliminating unnecessary complexities involving square roots within.

Algebraic Fractions

Algebraic fractions are expressions that contain variables in the numerator, denominator, or both. The exercise presented involves algebraic fractions where rationalizing the denominator is important to simplify the expression.The technique involves multiplying both the top and bottom of the fraction by a suitable radical that removes the square root from the denominator. For example:

• If you start with \( \frac{23}{5p^2\sqrt{2p}} \), multiply it by \( \frac{\sqrt{2p}}{\sqrt{2p}} \).
• This multiplication creates \( \frac{23 \sqrt{2p}}{5p^2 \times 2p} \), resulting in a numerator \( 23 \sqrt{2p} \) and a denominator \( 10p^3 \).

Once rationalized, the algebraic fraction \( \frac{23 \sqrt{2p}}{10p^3} \) is a cleaner form with no radicals in the denominator. This skill is crucial for solving complex rational equations efficiently.