Problem 90 - Intermediate Algebra

Question

Use rational expressions to write as a single radical expression. $$ \sqrt[6]{y} \cdot \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} $$

Step-by-Step Solution

Answer

The single radical expression is \( \sqrt[10]{y^9} \).

Step 1: Express Each Radical with Fractional Exponents

The first step in combining these radical expressions is to express each of them using fractional exponents. This allows us to manipulate them more easily.- The expression \( \sqrt[6]{y} \) can be rewritten as \( y^{1/6} \).- The expression \( \sqrt[3]{y} \) can be rewritten as \( y^{1/3} \).- The expression \( \sqrt[5]{y^2} \) can be rewritten as \( y^{2/5} \).

Step 2: Apply the Product of Powers Property

Now that each of the radicals is expressed as a power of \( y \), we can multiply these expressions using the property that states \( a^m \cdot a^n = a^{m+n} \).We will add the exponents:\[ y^{1/6} \cdot y^{1/3} \cdot y^{2/5} = y^{1/6 + 1/3 + 2/5} \]

Step 3: Find a Common Denominator and Add the Exponents

To add the exponents, we need a common denominator:- The denominators are 6, 3, and 5. The least common multiple (LCM) of these numbers is 30.- Convert each fraction: - \( \frac{1}{6} = \frac{5}{30} \) - \( \frac{1}{3} = \frac{10}{30} \) - \( \frac{2}{5} = \frac{12}{30} \)Add the exponents:\[ \frac{5}{30} + \frac{10}{30} + \frac{12}{30} = \frac{27}{30} \]

Step 4: Simplify the Exponent

The fraction \( \frac{27}{30} \) can be simplified.Find the greatest common divisor (GCD) of 27 and 30, which is 3, and simplify:\[ \frac{27}{30} = \frac{9}{10} \]Thus, the resulting exponent is \( \frac{9}{10} \).

Step 5: Write as a Single Radical Expression

Now that we have the final exponent \( \frac{9}{10} \), we can write the expression back as a radical expression:\[ y^{9/10} = \sqrt[10]{y^9} \] This represents all the original radicals combined into a single radical expression.

Key Concepts

Fractional Exponents

When you see numbers or variables under a radical symbol, we can express them using fractional exponents. This technique helps simplify and manipulate expressions.
Fractional exponents are another way to symbolize roots. Instead of using the radical sign, you use exponents that are fractions. - The denominator of the fraction indicates the type of root (square root, cube root, etc.).
- The numerator represents the power raised within that root.
For instance,

• \( \sqrt[n]{a} \) becomes \( a^{1/n} \)
• When there's an exponent inside, like \( \sqrt[n]{a^m} \), it converts to \( a^{m/n} \).

This conversion is extremely helpful in simplifying and solving expressions with multiple roots.

Product of Powers Property

The product of powers property is a useful algebraic rule that tells us how to handle when we multiply like bases with different exponents. When you multiply expressions that share the same base, you add their exponents together.
Here's the rule:
- For any real number \(a\) and any integers \(m\) and \(n\), it follows that: \[ a^m \cdot a^n = a^{m+n} \]
This property simplifies multiplying terms because it reduces it to basic addition of exponents. For example, multiplying \( y^{1/6} \), \( y^{1/3} \), and \( y^{2/5} \) results in:

• Adding the exponents: \(1/6 + 1/3 + 2/5\)

Using the product of powers property enables us to simplify complex expressions by focusing on the exponents.

Radical Expressions

Radical expressions are mathematical expressions that involve roots. Typically, they're written with the radical symbol (\( \sqrt{} \)), followed by some quantity.
Radicals can be square roots, cube roots, or any higher roots. Because of their complexity, converting radicals to use fractional exponents (as mentioned previously) makes them easier to handle.

• The square root \( \sqrt{a} \) implies \( a^{1/2} \)
• Cube root \( \sqrt[3]{a} \) is \( a^{1/3} \)
• Similarly, the sixth root \( \sqrt[6]{y} \) is expressed as \( y^{1/6} \)

By converting radical expressions into a common format using fractional exponents, we can simplify their manipulation and combine them more easily.

Common Denominator

When we work with fractional exponents, particularly in addition, finding a common denominator is crucial. Adding fractions involves making sure all the fractions have the same denominator.
This allows us to add the numerators directly.
For the expression \( y^{1/6} \cdot y^{1/3} \cdot y^{2/5} \), here’s how we find a common denominator:

• Identify the denominators: 6, 3, and 5
• Determine their Least Common Multiple (LCM), which is 30
• Convert each fraction:
  • \( \frac{1}{6} = \frac{5}{30} \)
  • \( \frac{1}{3} = \frac{10}{30} \)
  • \( \frac{2}{5} = \frac{12}{30} \)

With all exponents over the common denominator, we can sum them up and simplify the result efficiently, showcasing the beauty of combining fractional exponents.