Problem 50 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[3]{\frac{9}{16 p^{4}}}$$

Step-by-Step Solution

Answer

\( \frac{\sqrt[3]{3^2}}{2^{\frac{4}{3}} p^{\frac{4}{3}}} \) is simplified.

Step 1: Express the Radicand as a Product of Perfect Powers

The given problem is \( \sqrt[3]{\frac{9}{16 p^{4}}} \). To begin simplifying the expression, let's first express the components of the fraction under the cube root sign such that their roots can be easily determined. We rewrite 9 as \( 3^2 \) and 16 as \( 4^2 \) which is also \( (2^2)^2 = 2^4 \), and \( p^4 \) is already a perfect power.

Step 2: Apply the Cube Root Separately to the Numerator and Denominator

Now apply the cube root separately to the numerator and the denominator:\[\sqrt[3]{\frac{3^2}{2^4 p^{4}}} = \frac{\sqrt[3]{3^2}}{\sqrt[3]{2^4 p^{4}}}\]

Step 3: Simplify the Cube Root of the Denominator

Given the denominator, apply the properties of exponents:\[\sqrt[3]{2^4 p^{4}} = 2^{\frac{4}{3}} p^{\frac{4}{3}}\]There's no simplification for 2 \( \frac{4}{3} \) and for \( p \), it remains as is.

Step 4: Write the Expression in Simplified Form

The cube root of the numerator, \( \sqrt[3]{3^2} \), remains as is since 3 is not a perfect cube, and is written in simplified radical form.The simplified fraction is:\[\frac{\sqrt[3]{3^2}}{2^{\frac{4}{3}} p^{\frac{4}{3}}}\] This is the simplified form of the given expression.

Key Concepts

Radical Expressions

A radical expression involves roots, such as square roots, cube roots, and more. The expression is usually written with a radical symbol \( \sqrt{} \), which denotes the root to be extracted. In our exercise, we worked with a cube root \( \sqrt[3]{} \), which means we extract the number that, when multiplied by itself three times, will give the original number under the root.

When simplifying radical expressions, it's essential to ensure that all factors under the radical are simplified to their possible extent. This process often involves breaking down numbers or expressions into their prime factors or powers. For example, in simplifying \( \sqrt[3]{9} \), we express 9 as \( 3^2 \), using properties of exponents to re-write parts of the expression into simpler forms.

• Radicals can often be simplified by expressing the number underneath as a product of perfect powers.
• For cube roots, find if any factors are perfect cubes to simplify them easily.
• The goal is to reach the simplest radical form where no factor inside the radical can be extracted.

Mastering these steps makes working with radicals much smoother and enhances understanding of different root concepts.

Cube Roots

Cube roots provide a specific type of radical expression where the root is of degree three. This means you are finding a number that, when used in a product three times, yields the original number. With cube roots, like in the expression \( \sqrt[3]{\frac{9}{16 p^{4}}} \), we focus on determining what can be extracted from the radicand.

To simplify cube roots, consider whether the number or expression can be broken down into multiples or factors that are cubes. If they can be identified as such, you can take them out of the root, simplifying the expression. On other occasions, simplifying the cube root involves reducing complex numbers into their simpler form.

• Cubic factors: Identify parts of the number that are cubes, such as in the expression \( 8 = 2^3 \).
• Fractional radicands: Apply the cube root individually to the numerator and the denominator to see if they can be simplified.
• Cube roots are less common but understanding them improves handling higher degree radicals.

By recognizing and extracting cube factors, simplifying complex expressions becomes far less daunting for students.

Exponent Rules

Exponent rules help simplify expressions that involve powers, like those under radical signs. The rule \( a^{m/n} = \sqrt[n]{a^m} \) is crucial when dealing with radicals because it allows conversion between radical and exponential forms. For the given exercise, we applied these rules to simplify both the numerator and denominator.

Key rules include:

• Product of powers: \( a^m \times a^n = a^{m+n} \), combines powers with the same base.
• Power of a power: \( (a^m)^n = a^{m \times n} \), used when raising a power to another power.
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \) if \( a eq 0 \), governs division of same bases.
• Negative exponent: \( a^{-n} = \frac{1}{a^n} \), to rewrite power terms.
• Fractional exponents: Help rewrite roots; for cube roots, \( a^{1/3} = \sqrt[3]{a} \).

Understanding these rules is essential for manipulating and simplifying expressions with both radicals and exponents, ensuring mathematical tasks can be handled with ease.