Problem 73 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \frac{8^{\frac{1}{4}} a^{\frac{5}{6}} b^{\frac{3}{6}}}{\left(27 c^{4}\right)^{\frac{1}{6}}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{\sqrt[4]{8} \cdot \sqrt[6]{a^5} \cdot \sqrt{b}}{27^{\frac{1}{6}} c^{\frac{2}{3}}}\).

Step 1: Simplify the Denominator

First, simplify the expression in the denominator. Recognize that \((27 c^{4})^{\frac{1}{6}}\) can be rewritten by distributing the exponent, resulting in \(27^{\frac{1}{6}}\) and \((c^4)^{\frac{1}{6}}\). Therefore, the denominator becomes \(27^{\frac{1}{6}} c^{\frac{4}{6}}\). Simplifying \(c^{\frac{4}{6}}\) results in \(c^{\frac{2}{3}}\).

Step 2: Rewrite Numerator as Radicals

Next, rewrite each term in the numerator as a radical expression. The expression \(8^{\frac{1}{4}}\) can be written as \(\sqrt[4]{8}\). The term \(a^{\frac{5}{6}}\) becomes \(\sqrt[6]{a^5}\). The term \(b^{\frac{3}{6}}\) simplifies to \(b^{\frac{1}{2}} = \sqrt{b}\). Therefore, the numerator is \(\sqrt[4]{8} \cdot \sqrt[6]{a^5} \cdot \sqrt{b}\).

Step 3: Combine and Simplify the Expression

Now, combine the simplified numerator and denominator together. You get: \[\frac{\sqrt[4]{8} \cdot \sqrt[6]{a^5} \cdot \sqrt{b}}{27^{\frac{1}{6}} c^{\frac{2}{3}}}\]As a next step, you might consider expressing combined terms under one radical if possible, but this is a simple radical form given the initial requirements.

Key Concepts

Exponent Rules

When dealing with radical expressions that involve fractions and exponents, exponent rules become our best friend. These rules help us to break down and simplify expressions in radical form. Here are the key exponent rules you need to be familiar with:

• Product of Powers Rule: When multiplying two powers that have the same base, you add their exponents. For example, \[ a^m \cdot a^n = a^{m+n} \]
• Power of a Power Rule: When raising a power to another power, you multiply the exponents. For instance, \[(a^m)^n = a^{m\cdot n}\]
• Power of a Product Rule: Distributes an exponent over a multiplication. So, \[(ab)^n = a^n \cdot b^n\]

Applying these rules to simplify exponents is crucial in transitioning them into radical expressions.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. This means that no perfect power factors remain inside the radical, and the index is as small as possible. Understanding some basics will help:

• Prime Factorization: Decompose the number inside the radical into its prime factors. This can help simplify the radical.
• Look for Perfect Powers: Check if any of the numbers inside the radical are perfect squares (for square roots), perfect cubes (for cube roots), etc.
• Reduce Radicals: If a number has a factor that is a perfect power, take it out of the radical. For example, \[\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}\]

By understanding and applying these principles, you can effectively simplify radical expressions.

Numerator and Denominator

In any fraction, the numerator is the top part, and the denominator is the bottom part. When working with radical expressions that involve these fractions, simplifying each component is key:

• For the numerator, once each term is rewritten as a radical (as in our example), ensure it is in its simplest form.
• For the denominator, apply rules for exponentials and radicals to ensure it's expressed as simply as possible. Distributing powers across terms can often lead to simpler forms.

In our given expression, the numerator was simplified by converting each term to its respective radical form, while the denominator was carefully tackled by applying power distribution. This organized approach aids in overall simplification.

Rational Exponents

Rational exponents provide an alternative way of expressing roots and powers, combining both into a single expression. They follow this general format: \[a^{\frac{m}{n}}\]Where:

• Base \(a\) is the number we are working with.
• Numerator \(m\) is the power to which the base should be raised.
• Denominator \(n\) represents the root we are taking.

For example, \[8^{\frac{1}{4}} = \sqrt[4]{8}\]. This notation allows us to easily convert the expression between radical and exponential forms. It's a powerful tool in algebra for simplifying and solving equations in a unified and systematic manner.