Problem 597 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(64 s^{\frac{3}{5}}\right)^{\frac{1}{6}} $$

Step-by-Step Solution

Answer

2s^{\frac{1}{10}}

Step 1: Apply the power rule

Use the power rule \( (a^m)^n = a^{m\cdot n} \) to simplify the expression. Here, \(a = 64s^{\frac{3}{5}}\), \(m = \frac{1}{6}\). So, apply the rule: \((64s^{\frac{3}{5}})^{\frac{1}{6}} = 64^{\frac{1}{6}} \cdot \left(s^{\frac{3}{5}}\right)^{\frac{1}{6}}\).

Step 2: Simplify the constants

Simplify \(64^{\frac{1}{6}}\). Since \(64 = 2^6\), is the sixth root of 64 divided to end up with \(2\). Thus \(64^{\frac{1}{6}} = 2\).

Step 3: Simplify the variable expression

Now, simplify the variable expression \((s^{\frac{3}{5}})^{\frac{1}{6}}\) using the power rule: \((s^{\frac{3}{5}})^{\frac{1}{6}} = s^{\frac{3}{5}\cdot\frac{1}{6}} = s^{\frac{3}{30}} = s^{\frac{1}{10}}\).

Step 4: Multiply the simplified results

Combine the simplified expressions: \(64^{\frac{1}{6}} \cdot (s^{\frac{3}{5}})^{\frac{1}{6}} = 2 \cdot s^{\frac{1}{10}} = 2s^{\frac{1}{10}}\).

Key Concepts

Power Rule

The power rule is a fundamental concept in algebra that helps simplify expressions involving exponents. It states that when you raise a power to another power, you multiply the exponents.
Here's the rule: \( (a^m)^n = a^{m \times n} \).
In our exercise, this rule is applied to the expression \( (64 s^{3/5})^{1/6} \).
By using the power rule, we can separate the constants and the variable part, making the expression easier to handle. This saves time and reduces complexity in solving algebraic problems.

Fractional Exponents

Fractional exponents are another essential topic in algebra. They represent roots and provide a convenient way to express radicals. For example, \( a^{1/n} \) represents the nth root of a.
In our exercise, we encounter fractional exponents such as \( \frac{3}{5} \) and \( \frac{1}{6} \).
When simplifying \( (s^{3/5})^{1/6} \), we can use the power rule to multiply these exponents: \(\frac{3}{5} \times \frac{1}{6} = \frac{3}{30} = \frac{1}{10} \).
Understanding fractional exponents allows us to handle expressions involving roots and powers more effectively.

Properties of Exponents

The properties of exponents are key tools for simplifying algebraic expressions. These include:

• Product of Powers: \( a^m \times a^n = a^{m + n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{mn} \)
• Power of a Product: \( (ab)^m = a^m b^m \)

In our solution, we specifically use the 'power of a power' property. This helps us simplify \( (64 s^{3/5})^{1/6} \) step by step.
The properties of exponents are indispensable when dealing with more complex algebraic expressions.

Simplifying Radicals

Simplifying radicals involves expressing them in their simplest form. Often, radicals can be represented using fractional exponents.
For instance, the sixth root of \(64 \) is the same as \( 64^{1/6} \).
In our problem, \( 64^{1/6} = 2 \) because \( 64 = 2^6 \).
Another step involves simplifying \( (s^{3/5})^{1/6} \), which gives us \( s^{1/10} \).
By combining these results, we get the simplified form of the original expression: \( 2s^{1/10} \). Knowing how to simplify radicals is crucial for making complex algebraic problems more manageable.