Problem 65 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \frac{\sqrt{32}}{\sqrt{2}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{y^{1/5}}{x^{4/5}}\).

Step 1: Simplify expression inside the root

Rewrite the fifth root as a power to make it easier to simplify the expression. Using the rule \(\sqrt[n]{a^m} = a^{m/n}\), the expression can be rewritten as \(x^{-4/5} y^{1/5}\).

Step 2: Simplify negative power

The power -4/5 can be simplified by moving the term to the denominator part of the fraction, making the power positive. Hence, \(x^{-4/5} y^{1/5}\) simplifies to \(\frac{y^{1/5}}{x^{4/5}}\).

Step 3: Rationalize the denominator

As all variables are positive, the denominator cannot be rationalized in this case. Base numbers should be the same in order to get rational numbers in denominator. In this case, rationalize the denominator doesn't make any sense. So, our final simplified form is \(\frac{y^{1/5}}{x^{4/5}}\).

Key Concepts

Simplifying Radicals

Radicals often seem tricky, but they become a lot simpler once you understand the rules. In mathematics, a radical expression involves a root, such as a square root or a cube root. Here, we are dealing with a fifth root, also known as a radical of degree five. To simplify a radical, follow these steps:

• First, convert the radical expression into an expression using exponents. This makes it easier to manage and manipulate.
• The n-th root of a number can be expressed as a power: \(\sqrt[n]{a} = a^{1/n}\).

Using these steps, any expression of the form \(\sqrt[5]{x^{-4} y}\) can be rewritten with rational exponents as \(x^{-4/5} y^{1/5}\). It's much easier to perform operations on expressions in this form. Simplifying radicals allows you to work with expressions more comfortably, making complex calculations more manageable. Don't be intimidated by the roots; they are just another way of expressing powers.

Negative Exponents

Negative exponents can sometimes cause confusion, but they're just a reflection of division rather than multiplication. Here's how you deal with them:

• A term with a negative exponent can be rewritten as a reciprocal to make the exponent positive. For instance, \(x^{-n}\) is the same as \(\frac{1}{x^n}\).
• This rule helps in simplifying expressions since it allows the transformation of negative powers into a more straightforward form.

In our example, the expression \(x^{-4/5} y^{1/5}\) includes \(x\) raised to a negative power. By rewriting it, you bring the base \(x\) to the denominator: \(\frac{y^{1/5}}{x^{4/5}}\). Remember, a negative exponent signals that the base is on the wrong side of the fraction line and needs flipping. Hence, convert each negative exponent by taking the reciprocal, turning the exponent positive and making the expression simpler to understand and use in further calculations.

Rational Exponents

Rational exponents are exponents that are fractions, and they serve as an alternative way of expressing radicals. Here's how they function:

• The expression \(a^{m/n}\) is equal to \(\sqrt[n]{a^m}\), which means \(a\) raised to the power of the fraction \(m/n\) can be interpreted as the n-th root of \(a\) raised to the \(m\) power.
• This makes operations like multiplication and division more straightforward since it avoids the cumbersome radical notation.

For example, in the original expression, \(\sqrt[5]{x^{-4} y}\) is converted into expressions with rational exponents as \(x^{-4/5} y^{1/5}\). This conversion makes complicated radical expressions more manageable and paves the way for more straightforward arithmetic operations. Using rational exponents is a powerful tool, allowing you to express and deal with roots using a consistent form that integrates seamlessly with other algebraic laws.