Problem 31 - Intermediate Algebra

Question

Rationalize each denominator. See Examples 1 through 3. $$ \sqrt[4]{\frac{16}{9 x^{7}}} $$

Step-by-Step Solution

Answer

\( \frac{2\sqrt[4]{9x}}{3x} \)

Step 1: Rewrite the expression with roots

Start by rewriting the given fourth root in terms of entries under the root expression. The expression given is \( \sqrt[4]{\frac{16}{9x^7}} \). This can be expressed as \( \left(\frac{16}{9x^7}\right)^{\frac{1}{4}} \).

Step 2: Separate the root to the numerator and the denominator

Using the property \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \), separate the fourth root: \( \frac{\sqrt[4]{16}}{\sqrt[4]{9x^7}} \).

Step 3: Simplify the numerator

Simplify \( \sqrt[4]{16} \). Since \( 16 = 2^4 \), \( \sqrt[4]{16} = 2 \).

Step 4: Analyze the denominator terms

Look at the denominator \( \sqrt[4]{9x^7} \), which simplifies to \( \left(9x^7\right)^{\frac{1}{4}} \) or \( 9^{\frac{1}{4}} \times x^{\frac{7}{4}} \). Notice that \( 9 = 3^2 \), so \( 9^{\frac{1}{4}} = 3^{\frac{1}{2}} = \sqrt{3} \).

Step 5: Rationalize the denominator

To rationalize \( \frac{1}{\sqrt[4]{9x^7}} \), multiply both the numerator and the denominator by \( x^{1/4} \) and \( 3^{1/4} \) so that the powers of \( x \) and \( 3 \) become rationalized: \( \frac{2 \times \sqrt[4]{9x} }{ \left(3x^{7/4}\right)^{1/4} \times 3^{3/4}x^{3/4} } = \frac{2\sqrt[4]{9x}}{3\sqrt[4]{x}} \).

Step 6: Simplify the expression

After multiplying and rationalizing the expression, simplify further to reach the final form. The denominator \( \left(3x^{7/4}\right)^{1/4} \times 3^{1/4}x^{1/2}\) can split as \( \frac{2\sqrt[4]{9}\sqrt[4]{x} }{3x} \).

Step 7: Final answer

The final simplified rationalized expression is \( \frac{2\sqrt[4]{9x}}{3x} \).

Key Concepts

Fourth Roots

In mathematics, when we talk about fourth roots, we are focusing on finding a number that, when raised to the power of four, gives us the original number. In our exercise, we see \(\sqrt[4]{16}\).
This expression seeks a number that when raised to four equals 16.
To compute this in practice, try writing the base number in a form that easily translates to the fourth power.

• For example, \(16\) is the same as \(2^4\).
• Thus, \(\sqrt[4]{16} = 2\), because \(2^4 = 16\).

Breaking it down into smaller steps can make finding fourth roots less daunting. Similarly, fourth roots can be separated in fractions, leading to easier computation.

Simplifying Expressions

Simplifying expressions involves rewriting them in their simplest or most efficient form. This includes reducing numbers, breaking down fractions, or using exponent rules. The process is about making an expression easier to work with or understand.

In the provided exercise, simplifying occurs in several stages:

• First, the expression is split into numerical parts and variable parts.
• Both the numerator and denominator take separate routes in simplification.
• The numerical component \(\sqrt[4]{16}\) turns into \(2\) for the numerator.
• The denominator ends up in the form \( \sqrt[4]{9x^7}\), where the numbers are simplified to \( \sqrt[4]{9} \times x^{7/4}\).

Each step ensures the end expression is less complex than the starting one, aiding further operations or rationalizing.

Exponents

Exponents are a mathematical method used to represent repeated multiplication. For instance, \(x^7\) means multiplying x by itself seven times. Exponents also act as a powerful tool when simplifying expressions, especially those involving roots.

In this exercise, recognizing how exponents work with roots is crucial:

• The expression \((9x^7)^{1/4}\) uses a fractional exponent to denote the fourth root.
• Breaking it down, you have \(9^{1/4} \times x^{7/4}\).
• A fractional exponent like \(\frac{1}{4}\) indicates taking a fourth root.

Moreover, understanding that you can alter the position of numbers linked to exponents helps in other operations, like rationalizing denominators, which transforms irrational expressions into rational ones. The end result is a neat, simplified representation of the original problem.