Problem 29 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{5 a}{\sqrt[5]{8 a^{9} b^{11}}}\)

Step-by-Step Solution

Answer

The rationalized form is \(\frac{5b}{2a}\).

Step 1: Understand the Problem

To rationalize the denominator of the given expression \( \frac{5a}{\sqrt[5]{8a^9b^{11}}} \), we need to remove the radical sign by converting the expression to have integer exponents in the denominator.

Step 2: Identify the Needed Powers

The denominator is \( \sqrt[5]{8a^9b^{11}} \). Observe that to rationalize, we need each term within the radical to be a fifth power. Hence, we need \((8a^9b^{11})^{1/5}\) to become \((8a^9b^{11})^{5/5}\).

Step 3: Find the Missing Powers

We need to multiply the inside of the fifth root such that \(8a^9b^{11}\) becomes \((8a^9b^{11})^1\). This means we need to multiply by enough to make the power 5. So, we multiply by \(a\) and \(b^4\) because this gives us \(a^{9+1}\) and \(b^{11+4}\) which are multiples of 5.

Step 4: Calculate the New Expression

Multiply numerator and denominator by \(a^1b^4\) to eliminate the radical:\[\frac{5a}{\sqrt[5]{8a^9b^{11}}} \times \frac{a^1b^4}{a^1b^4} = \frac{5ab^4}{\sqrt[5]{8a^9b^{11}a^1b^4}} = \frac{5ab^4}{\sqrt[5]{8a^{10}b^{15}}}\]

Step 5: Simplify the Expression

Now, rationalize the denominator by converting it entirely:\[\frac{5ab^4}{\sqrt[5]{8a^{10}b^{15}}} = \frac{5ab^4}{(8a^{10}b^{15})^{1/5}} = \frac{5ab^4}{2a^2b^3}\] Here, \((8)^{1/5}=2\), \((a^{10})^{1/5}=a^2\), and \((b^{15})^{1/5}=b^3\).

Step 6: Simplify the Fraction

After rationalization, simplify the fraction \[\frac{5ab^4}{2a^2b^3} = \frac{5b}{2a}\]by canceling terms in the numerator and the denominator: \(a\) from the numerator cancels with \(a^2\) leaving \(a\) in the denominator, and \(b^3\) cancels with \(b^4\) leaving \(b\) in the numerator.

Key Concepts

Radical Expressions

Radical expressions involve roots such as square roots, cube roots, and more generally, nth roots. These expressions contain a radical symbol, which resembles a checkmark (√), followed by a number or expression underneath called the radicand. In our case, we're dealing with a fifth root denoted as \( \sqrt[5]{...} \).

• The goal with radical expressions often involves simplifying them or converting them into a form without a radical, especially when they appear in denominators.
• This process is called rationalizing the denominator, where we aim to eliminate the root by employing properties of exponents.

Radical expressions can complicate algebraic manipulations, especially when combined with variables or coefficients. Interpret them as expressions under fractional powers, like \( a^{1/2} \) for a square root. Understanding these as exponents helps simplify or manipulate such expressions easily.

Fifth Roots

Fifth roots are a specific type of radical expression where the index of the root is five. To rationalize or handle expressions with fifth roots, you need certain skills:

• You seek to express the radicand in terms that can be rewritten with integer exponents. For instance, \( \sqrt[5]{x} = x^{1/5} \).
• When rationalizing, you aim for a complete fifth power as the exponent in the denominator. This often involves multiplication to achieve the needed power.

In our example, the radicand was initially \(8a^9b^{11}\). To rationalize, you adjust the powers by identifying and multiplying by terms needed to reach multiples of five: for \(a\) and \(b\), this involved multiplying by \(a\) and \(b^4\). This strategy relies on multiplying by expressions that, when combined with the existing terms, complete the power requirement to make each a full fifth power.

Simplifying Fractions

Simplifying fractions is a key concept in algebra. When faced with complex expressions, breaking them down to simpler terms is crucial to ease understanding and further computation. Here's how you do it:

• You identify and divide out common factors from the numerator and the denominator. For example, \( \frac{4x}{2x} \) simplifies to \( \frac{2}{1} = 2 \).
• Once the expression is rationalized, like in our exercise, further simplification may involve canceling like terms across the division line.

After rationalizing the expression \( \frac{5ab^4}{2a^2b^3} \), it simplifies by canceling the common factors of \(a\) and \(b\). You are left with \( \frac{5b}{2a} \), which is easier and more efficient to work with. Simplified expressions are crucial for clearer results and simpler calculations in algebra.

Integer Exponents

Integer exponents are exponent values that are whole numbers. When dealing with roots like fifth roots, expressing terms with integer exponents simplifies algebraic manipulation. Here's why they're important:

• An integer exponent shows repeated multiplication, such as \(x^3 = x \times x \times x\).
• Working with integers makes it easier to identify common bases and apply exponent arithmetic laws, such as \(x^a \times x^b = x^{a+b}\).

In the denominator, transitioning from a radical expression like \( (8a^{10}b^{15})^{1/5} \) to integer exponents as \( 2a^2b^3 \) simplifies division and helps achieve the goal of rationalization. This approach ensures that calculations don't involve radicals, making them straightforward. Overall, using integer exponents supports efficient and accurate algebraic operations.