Problem 66 - Algebra A Combined Function

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[8]{4 y^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4^{1/8}y^{1/4} \).

Step 1: Express the Radical as a Power

To use rational exponents, convert the radical expression \( \sqrt[8]{4y^2} \) to a power. The expression \( \sqrt[n]{a} \) can be represented as \( a^{1/n} \). Thus, \( \sqrt[8]{4y^2} = (4y^2)^{1/8} \).

Step 2: Apply Power to Each Term

Distribute the exponent \( \frac{1}{8} \) to each term inside the parentheses: \( (4y^2)^{1/8} = 4^{1/8} \cdot (y^2)^{1/8} \). Now we can simplify each term separately.

Step 3: Simplify Each Term

Simplify the expression for each component with rational exponents:- For \( 4^{1/8} \), we leave it as is since it is already expressed as a power.- For \( (y^2)^{1/8} \), use the power of a power property: \( (y^2)^{1/8} = y^{2 \times \frac{1}{8}} = y^{1/4} \).

Step 4: Combine the Simplified Terms

Combine the simplified components to express the expression in a simpler form: \[ 4^{1/8} \cdot y^{1/4}. \] This is the simplified form using rational exponents.

Key Concepts

Simplifying Radicals

Simplifying radicals is a technique to make radical expressions easier to work with. A radical is an expression that includes a root, such as a square root or cube root. In many cases, simplifying a radical involves converting it into a form with rational exponents. This simplifies calculations and algebraic manipulations.

• Simplifying radicals often involves finding perfect squares or cubes within the expression.
• The goal is to rewrite the radical in its simplest form.
• It often involves factoring out numbers or variables that can be taken out from under the radical sign.

Understanding the conversion to rational exponents is key in simplifying radicals, as it leverages exponent rules to manipulate the expression more easily. This is particularly useful in advanced algebra where radicals might make direct calculation cumbersome.

Exponent Rules

Exponent rules are fundamental tools for simplifying expressions, especially when dealing with rational exponents. These rules guide how we handle operations involving exponents, such as multiplication and division.


Key exponent rules include:

• Product rule: When multiplying like bases, add their exponents: \( a^m \cdot a^n = a^{m+n} \).
• Quotient rule: When dividing like bases, subtract their exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a power rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).

When simplifying expressions with rational exponents, these rules become even more valuable in breaking down the problem systematically. For instance, in our problem, we apply the power of a power rule to manage the term \((y^2)^{1/8}\). This rule simplifies complex exponents efficiently by reducing them to a single, workable power.

Radical Expressions

Radical expressions are mathematical phrases that involve roots, such as square roots \( \sqrt{} \), cube roots \( \sqrt[3]{} \), and others like fourth or eighth roots. These expressions can initially seem daunting due to their complex appearance.

To handle radical expressions effectively, converting them into rational exponents is a useful strategy:

• This involves turning the root into an exponent, providing a familiar form to work with using exponent rules.
• For example, a cube root can be expressed as an exponent of \( \frac{1}{3} \), simplifying the manipulation process due to familiarity with exponent rules.
• In our exercise, \( \sqrt[8]{4y^2} \) was transformed into \((4y^2)^{1/8}\), making it easier to apply exponent rules directly.

Radical expressions, when converted, allow algebraic operations like multiplication, division, or raising powers to be executed more smoothly, helping students tackle complex algebra problems with ease. Expressing radicals as powers provides clarity and simplicity, crucial for higher-level math skills.