Problem 66 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{\sqrt{62}}{\sqrt{6}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( \sqrt[6]{\frac{y^{-3}}{x^{-4}}} \) is \( x^{\frac{2}{3}} \cdot \frac{1}{y^{\frac{1}{2}}} \)

Step 1: Convert the Root to an Exponent

The expression given is \( \sqrt[6]{\frac{y^{-3}}{x^{-4}}} \). The square root of \( \frac{y^{-3}}{x^{-4}} \) to the power of 6 is equal to \( (\frac{y^{-3}}{x^{-4}}) ^ {\frac{1}{6}} \).

Step 2: Simplify the Exponential Expression

Using exponent rules, simplify the expression to give \( (\frac{1/y^3}{1/x^4})^{\frac{1}{6}} \) which equates to \( (\frac{x^4}{y^3})^{\frac{1}{6}} \).

Step 3: Distribute the Exponent

Next, apply the exponent \( \frac{1}{6} \) to both \( x^4 \) and \( y^3 \) based on the rule \( (a\cdot b)^n = a^n \cdot b^n \) to get the simplified form \( x^{\frac{4}{6}} \cdot y^{-\frac{3}{6}} \)

Step 4: Simplify the Fractional Exponents

Fractional exponent \( x^{\frac{4}{6}} \) simplifies to \( x^{\frac{2}{3}} \), and \( y^{-\frac{3}{6}} \) simplifies to \( y^{-\frac{1}{2}} \)

Step 5: Rationalize the Denominator

Rationalizing the denominator means rewriting the expression so that there are no roots (or fractional powers) of numbers and variables in the denominator. Since \( a^{-n} = \frac{1}{a^n} \), \( y^{-\frac{1}{2}} \) can be rewritten as \( \frac{1}{y^{\frac{1}{2}}} \), this gives the final simplified form of the expression as \( x^{\frac{2}{3}} \cdot \frac{1}{y^{\frac{1}{2}}} \)

Key Concepts

Simplifying Exponential Expressions

Exponential expressions can sometimes look complicated, but by understanding the basic rules, you can simplify them easily. Let's break it down with a focus on the expression\[\left(\frac{x^4}{y^3}\right)^{\frac{1}{6}}\]. First, remember that when you have a fraction raised to an exponent, you need to distribute the exponent to both the numerator and the denominator.

• This means applying the exponent \(\frac{1}{6}\) to both \(x^4\) and \(y^3\).
• As a result, you get \(x^{\frac{4}{6}}\) and \(y^{\frac{3}{6}}\).

Once you have distributed the exponent, you can further simplify each term by reducing the fractions in the exponents. This step clarifies the expression and prepares it for further simplifications, like rationalization, if necessary.In our example, \(x^{\frac{4}{6}}\) simplifies to \(x^{\frac{2}{3}}\) and \(y^{\frac{3}{6}}\) simplifies to \(y^{\frac{1}{2}}\). With these simplifications complete, you are well on your way to having a clearer expression ready for rationalization.

Fractional Exponents

A fractional exponent is another way to represent roots, and simplifying them often involves converting the root to an exponent. In our exercise, after converting the root operation to fractional exponents, the expression becomes\[x^{\frac{4}{6}} \cdot y^{\frac{-3}{6}}\].The numerator of the fraction represents the power, while the denominator represents the root:

• For example, \(x^{\frac{4}{6}}\) can be interpreted as \(\sqrt[6]{x^4}\).
• Similarly, \(y^{\frac{-3}{6}}\) indicates a sixth root raised to a power of \(-3\).

Simplifying fractional exponents involves reducing these fractions. This means that the top number, or numerator, and the bottom number, or denominator, of the exponent can often be divided by a common factor to make the expression simpler. Reducing these exponents simplifies the expression, showing clearly the power and the root, making further operations more straightforward. In practice, reducing the fractions \(\frac{4}{6}\) to \(\frac{2}{3}\) and \(\frac{-3}{6}\) to \(-\frac{1}{2}\) allows you to express the original root operation in a much clearer way.

Root to Exponent Conversion

Converting roots to exponents is a powerful technique in algebra that helps simplify complex expressions. When faced with an expression like\[\sqrt[6]{\frac{y^{-3}}{x^{-4}}}\],realize that you can transform it to an exponential form for easier manipulation:

• In general, \(\sqrt[n]{a} = a^{\frac{1}{n}}\).
• Therefore, the sixth root of \(\frac{y^{-3}}{x^{-4}}\) becomes \((\frac{y^{-3}}{x^{-4}})^{\frac{1}{6}}\).

This conversion allows you to break down the problem into smaller parts that are often easier to handle. By expressing roots as exponents, you grant yourself access to a toolkit of exponent rules that make simplifying possible.When converted, exponent rules such as power of a power \((a^m)^n = a^{mn}\) can be used to distribute exponents over products or quotients, allowing for systematic simplification. In this case, converting the root enabled smoothing through exponent manipulation to unparalleled clarity in problem-solving.