Problem 73 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( \sqrt[15]{y^{11}} \).

Step 1: Express Each Radical Using Rational Exponents

First, we express each radical expression using rational exponents. The given expression \( \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} \) can be transformed as follows:1. \( \sqrt[3]{y} = y^{1/3} \)2. \( \sqrt[5]{y^{2}} = y^{2/5} \)

Step 2: Use Product of Powers Property

When multiplying powers with the same base, we add their exponents. The expression becomes:\[ y^{1/3} \cdot y^{2/5} = y^{1/3 + 2/5} \]

Step 3: Find a Common Denominator

To add the exponents \(1/3\) and \(2/5\), we need a common denominator. The least common multiple of 3 and 5 is 15.- Convert \(1/3\) to \(5/15\)- Convert \(2/5\) to \(6/15\)

Step 4: Add the Exponents

Now, add the fractions with the common denominator:\[\frac{5}{15} + \frac{6}{15} = \frac{11}{15}\]Thus, the expression simplifies to:\(y^{11/15}\)

Step 5: Write as a Single Radical Expression

Finally, convert the rational exponent back into a radical expression:- The base is \(y\).- The numerator of the exponent (11) is the power to which \(y\) is raised.- The denominator of the exponent (15) indicates the root.Thus, the expression is \( \sqrt[15]{y^{11}} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and others. When we say "square root," we mean an expression like \(\sqrt{x}\), which implies finding a number that multiplies by itself to give \(x\). With cube roots, like \(\sqrt[3]{x}\), we are looking for a number that, when used three times in multiplication, results in \(x\). Each of these radical expressions can be rewritten for simplicity, particularly when we want to perform operations like multiplication or division.

• Square Root: \(\sqrt{x} = x^{1/2}\)
• Cube Root: \(\sqrt[3]{x} = x^{1/3}\)
• Fifth Root: \(\sqrt[5]{x} = x^{1/5}\)

Converting radicals to rational exponents often makes it easier to manage and combine these expressions, particularly when we need to perform calculations that involve addition or multiplication of exponents.

Rational Exponents

Rational exponents express roots as fractional exponents, which allow for easier manipulation of mathematical expressions. For example, an expression like \(\sqrt{x}\) in radical form is expressed as \(x^{1/2}\) in rational exponent form. This transformation leverages the rules of exponents, allowing us to treat roots as fractions and manage them with algebraic operations easily.Some examples include:

• \(\sqrt[3]{y} = y^{1/3}\), which is useful when moving between radicals and polynomial expressions.
• \(\sqrt[5]{y^2} = y^{2/5}\), showcasing how both powers and roots can be expressed together in a single, simplified expression.

Converting between these forms helps streamline operations like multiplication and division of expressions with the same base.

Product of Powers

The product of powers property is a core principle in simplifying expressions with like bases. When you multiply powers with the same base, such as \(a^m\) and \(a^n\), you add their exponents. For instance, the expression \(y^{1/3} \cdot y^{2/5}\) becomes \(y^{1/3 + 2/5}\).This is because:

• The base \(y\) remains consistent.
• The operation focuses solely on combining the exponents through addition.

The combined expression still has the same base, but now with an exponent that is the sum of the individual powers, allowing for clearer and unified expression.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. This concept is particularly crucial when dealing with rational exponents, as seen in manipulating expressions like \(y^{1/3 + 2/5}\). To add these fractions, we found a common denominator, which is the least common multiple of the denominators.

• The least common multiple of 3 and 5 is 15.
• Convert each fraction: \(1/3\) becomes \(5/15\) and \(2/5\) becomes \(6/15\).
• Add the converted fractions: \(5/15 + 6/15 = 11/15\).

Using a common denominator ensures the fractions have the same base, allowing for straightforward addition, leading to a more tractable mathematical expression, like converting back into one radical form: \(\sqrt[15]{y^{11}}\).