Problem 74 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \sqrt[5]{x^{3} y^{2}} \sqrt[10]{x^{4} y^{16}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x y^{2} \).

Step 1: Rewrite Using Rational Exponents

Rewrite both expressions using rational exponents. The expression \( \sqrt[5]{x^{3} y^{2}} \) can be written as \( (x^3 y^2)^{1/5} \), and \( \sqrt[10]{x^{4} y^{16}} \) can be written as \( (x^4 y^{16})^{1/10} \).

Step 2: Apply Power Rule for Exponents

Apply the power rule: \((a^m)^n = a^{m \cdot n}\). For \( (x^3 y^2)^{1/5} \), apply the rule to get \( x^{3/5} y^{2/5} \). For \( (x^4 y^{16})^{1/10} \), apply the rule to get \( x^{4/10} y^{16/10} = x^{2/5} y^{8/5} \).

Step 3: Combine the Terms

Multiply the expressions by adding the exponents corresponding to the same base. For \( x \): \( x^{3/5} \cdot x^{2/5} = x^{(3/5 + 2/5)} = x^{1} \). For \( y \): \( y^{2/5} \cdot y^{8/5} = y^{(2/5 + 8/5)} = y^{2} \).

Step 4: Simplify the Combined Expression

Combine the results to get the final simplified expression. Therefore, the result is \( x^{1} y^{2} \) or simply \( x y^{2} \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves rewriting them in a form that is easier to understand or solve. In this exercise, we are given two radical expressions that need to be combined into a simpler expression using rational exponents. The initial form features roots, which are not as straightforward for manipulation as exponents.

• Begin by transforming roots into exponents. This makes it easier to manipulate using exponent rules.
• Recall that the root of a term, such as the fifth or tenth root, can be expressed as an exponent: for the fifth root, write as a power of 1/5, and for the tenth root, use a power of 1/10.

Once you have the rational exponent form, you can begin to apply further exponent rules to simplify the expression. The goal here is to have a single, clear expression that represents the original radicals effectively.

Power Rule for Exponents

The power rule for exponents is a vital tool when dealing with expressions involving powers. The rule states, if you have \[ (a^m)^n = a^{m \cdot n} \]Applying this rule allows you to manage complex exponents more effortlessly. In the exercise, we used this rule to alter expressions with rational exponents.

• Each part of the expression, such as \( (x^3 y^2)^{1/5} \), can be broken down using the power rule. This simplifies each base by multiplying the exponents.
• For instance, \[ (x^3)^{1/5} = x^{3/5} \] and similarly for other parts of the expression.

Understanding this rule allows you to reduce complex expressions by converting power-to-power situations into simpler single power expressions.

Rational Expressions

Rational expressions involve fractions in the exponents and are key to simplifying roots into exponents easily. This transformation is particularly handy since exponent rules are often easier to work with than roots.

• Use prime factorization of the root's index as a portion of the exponent when converting roots to rational expressions.
• Combining rational expressions in this exercise required adding exponents with the same base: \( x^{3/5} \cdot x^{2/5} = x^{1} \), adding the fractions \(3/5 + 2/5 = 1\).

By changing radical notation into rational exponent notation, you gain the ability to use all the properties of exponents to simplify expressions. This results in a more concise and manageable mathematical expression.