Problem 78 - Algebra A Combined Function

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

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

Step 1: Convert Radicals to Exponents

Convert each radical expression into exponential form using the property that \( \sqrt[n]{a} = a^{1/n} \). For the given radicals: 1. \( \sqrt[6]{y} = y^{1/6} \)2. \( \sqrt[3]{y} = y^{1/3} \)3. \( \sqrt[5]{y^2} = (y^2)^{1/5} = y^{2/5} \)

Step 2: Multiply the Exponents

Using the property of exponents that \( a^m \cdot a^n = a^{m+n} \), combine the expressions:\[ y^{1/6} \cdot y^{1/3} \cdot y^{2/5} = y^{1/6 + 1/3 + 2/5} \]

Step 3: Find a Common Denominator for Addition

To add these fractions, first find a common denominator. The denominators 6, 3, and 5 have a least common multiple of 30.Convert each fraction:- \( \frac{1}{6} = \frac{5}{30} \)- \( \frac{1}{3} = \frac{10}{30} \)- \( \frac{2}{5} = \frac{12}{30} \)

Step 4: Add the Exponents

Add the fractions together to get:\[ \frac{1}{6} + \frac{1}{3} + \frac{2}{5} = \frac{5}{30} + \frac{10}{30} + \frac{12}{30} = \frac{27}{30} \]

Step 5: Simplify the Exponent

Simplify the exponent by reducing the fraction:\[ \frac{27}{30} = \frac{9}{10} \]

Step 6: Write as a Single Radical Expression

Convert the expression with the simplified exponent back into a radical form:\[ y^{9/10} = \sqrt[10]{y^9} \]

Key Concepts

radicals

Radicals are expressions that include a root, such as a square root, cube root, or any higher-order root. When you see a radical symbol \( \sqrt[n]{a} \), it represents the \( n \)-th root of \( a \). A square root is the most common radical and is typically written without the index, while higher roots specify the index like \( \sqrt[3]{a} \) for a cube root.

To handle radicals, you often need to convert them into exponent form because it makes the math easier. For instance, \( \sqrt[3]{y} \) is equivalent to \( y^{1/3} \), because the cube root is just raising \( y \) to the power of \( \frac{1}{3} \). This conversion is crucial when you need to combine or simplify radicals as it allows you to leverage the properties of exponents.

Understanding that every radical has an equivalent exponent form helps in combining and simplifying expressions with different radicals. It's like finding a common language that they all speak, making calculations much simpler.

exponents

Exponents represent power or the number of times a number (called the base) is multiplied by itself. The expression \( a^n \) means that \( a \) is multiplied by itself \( n \) times.

A crucial rule of exponents is that when you multiply similar bases \( a^m \) and \( a^n \), you can simply add the exponents: \( a^{m+n} \). This rule is essential when you convert a product of radicals into exponent form, allowing you to simplify the expression in one step.

Understanding how to manipulate expressions with exponents, including multi-step operations like converting radicals to exponents, is an essential skill. It makes it easier to handle complex expressions and is particularly useful when working with algebraic expressions that contain multiple terms or when your goal is to simplify expressions to their most succinct form.

common denominators

When adding or subtracting fractions, like \( \frac{1}{6} \), \( \frac{1}{3} \), and \( \frac{2}{5} \), a common denominator is crucial. A common denominator is a shared multiple of all the denominators involved in your fractions.

To find it, first identify the least common multiple (LCM) of the denominators. For instance, with 6, 3, and 5, the LCM is 30. Next, convert each fraction to have this common denominator:

• \( \frac{1}{6} = \frac{5}{30} \)
• \( \frac{1}{3} = \frac{10}{30} \)
• \( \frac{2}{5} = \frac{12}{30} \)

By converting to a common denominator, adding fractions becomes a simple task of adding the numerators together. This method makes calculations straightforward and ensures accuracy in your results. Finding and using a common denominator is an indispensable tool in mathematics, especially when simplifying complex expressions.

simplifying fractions

Simplifying fractions involves reducing them to their simplest form, so that both the numerator and the denominator are as small as possible while maintaining the same value. For example, the fraction \( \frac{27}{30} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).

To simplify \( \frac{27}{30} \), identify the GCD of 27 and 30, which is 3. Divide both the numerator and the denominator by 3:

• \( \frac{27}{30} \rightarrow \frac{27 \div 3}{30 \div 3} = \frac{9}{10} \)

Simplifying fractions is important because it provides the cleanest and most understandable version of an expression. It also helps in comparing fractions and performing further arithmetic operations with ease. In algebra, simplifying fractions can often lead to more meaningful insights into the problem at hand, making them easier to solve or interpret.