Problem 58 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive and write your answer in radical notation. $$ \sqrt[3]{16} \cdot \sqrt{2} $$

Step-by-Step Solution

Answer

Simplified expression is \( \sqrt[6]{2^{11}} \).

Step 1: Express Radicals as Powers

To simplify the expression, start by rewriting each radical expression as a power. Recall that \( \sqrt[3]{a} = a^{1/3} \) and \( \sqrt{b} = b^{1/2} \).Therefore, our expression becomes: \[ \sqrt[3]{16} \cdot \sqrt{2} = 16^{1/3} \cdot 2^{1/2} \]

Step 2: Combine the Powers of Same Base

Since both bases are not the same, the expression cannot be directly combined using the properties of exponents. However, notice the relation: \( 16 = 2^4 \), thus, rewrite \( 16^{1/3} \) as \( (2^4)^{1/3} \).Substitute this back, so the expression becomes: \[ (2^4)^{1/3} \cdot 2^{1/2} \]

Step 3: Apply Exponent Rules

Now apply the rule \( (a^m)^n = a^{mn} \) to each term with the same base:\[ (2^4)^{1/3} = 2^{4 \cdot (1/3)} = 2^{4/3} \]Our expression inside the square root becomes: \[ 2^{4/3} \cdot 2^{1/2} \]

Step 4: Sum the Exponents

When multiplying terms with the same base, the exponents are added:\[ 2^{4/3} \cdot 2^{1/2} = 2^{(4/3 + 1/2)} \]We need a common denominator to add these fractions. The common denominator for \(3\) and \(2\) is \(6\):Convert the fractions: \(4/3 = 8/6\) and \(1/2 = 3/6\).Thus, the sum is: \[ 8/6 + 3/6 = 11/6 \]

Step 5: Simplify the Expression

Combine the exponents into the single power of two:\[ 2^{11/6} \]To write this in radical notation, split into two parts:\[ 2^{11/6} = (2^1)^{11/6} = \left( (2^1)^{1/6} \right)^{11} \]Thus, in radical form, we can express it as:\[ \sqrt[6]{2^{11}} \]

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when simplifying expressions with powers. When dealing with powers, several key rules help us manipulate the expressions effectively. Here are some essential exponent rules:

• Power of a Power Rule: When you have a base raised to a power and then that whole expression raised again to another power, like \( (a^m)^n \), you multiply the exponents. This results in \( a^{m \times n} \).
• Product of Powers Rule: When you multiply two expressions with the same base, you add their exponents, such as \( a^m \cdot a^n = a^{m+n} \).
• Quotient of Powers Rule: When you divide two expressions with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Zero Exponent Rule: Any nonzero number raised to the power of zero is 1: \( a^0 = 1 \).

By following these rules, you can simplify complex expressions by either reducing them or changing their forms. In this exercise, the power of a power and product of powers rules were used to combine and simplify the expression.

Simplifying Expressions

Simplifying expressions often involves converting complex notations into standard forms, making them easier to work with. In this example, the radicals were first expressed as fractional exponents, which is a key step. Simplifying using fractional exponents often involves these steps:
- Rewrite Radicals: Convert radicals to their exponential form, for example, \(\sqrt{a} = a^{1/2}\) and \(\sqrt[3]{a} = a^{1/3}\).
- Combine Like Terms: Look for terms with the same base to combine them using exponent rules.
In the given solution, after rewriting the radicals, we identified common bases. By expressing \(16\) as \(2^4\), we opened up the opportunity to apply the power of a power rule. This step simplified the expression significantly.
Moreover, always ensure variables remain positive if specified since it influences the simplification and final notations.

Fractional Exponents

Fractional exponents offer a different way to express radical expressions, making them easier to manipulate using algebraic rules. Each fraction's numerator and denominator have distinct meanings:

• The numerator of the fractional exponent represents the power to which the base is raised.
• The denominator indicates the root of the base, equivalent to the radical degree.

For instance, \( a^{m/n} \) means "take the nth root of a and then raise it to the mth power." Conversely, you could first apply the power and then the root, as these operations are interchangeable due to their mathematically equivalent nature.
Using fractional exponents in this exercise allowed continuous manipulation via exponent rules without returning to radical notation until the final step, ensuring clarity and ease of operation. Understanding this concept thoroughly conveys flexibility in solving and simplifying complex algebraic expressions.