Problem 68 - Algebra A Combined Function

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[8]{(y+1)^{4}} $$

Step-by-Step Solution

Answer

The simplified form is \( \sqrt{y+1} \).

Step 1: Rewrite the Radical as an Exponent

The expression \( \sqrt[8]{(y+1)^4} \) can be rewritten using rational exponents. A radical in the form of \( \sqrt[n]{a^m} \) is equivalent to \( a^{m/n} \). Here, \( (y+1)^4 \) is under an 8th root, so we rewrite it as \( (y+1)^{4/8} \).

Step 2: Simplify the Exponent Fraction

Next, simplify the fraction \( \frac{4}{8} \). This fraction can be reduced by dividing both the numerator and the denominator by their greatest common divisor, which is 4. So, \( \frac{4}{8} = \frac{1}{2} \). Thus, the expression \( (y+1)^{4/8} \) simplifies to \( (y+1)^{1/2} \).

Step 3: Express as a Simplified Radical

Finally, note that an exponent of \( \frac{1}{2} \) corresponds to the square root. Therefore, \( (y+1)^{1/2} \) is equivalent to \( \sqrt{y+1} \). The original radical \( \sqrt[8]{(y+1)^4} \) is simplified to \( \sqrt{y+1} \).

Key Concepts

Simplifying Radicals

When we talk about simplifying radicals, we mean taking a complex root expression and converting it into a simpler form. For instance, the original exercise begins with the 8th root of \(y+1\) raised to the 4th power. We simplify this by using the properties of exponents and understanding that radicals can be written in terms of fractional exponents.
It's important to first recognize the expression as a radical and then change it to an exponent with a fraction. For example, \(\sqrt[8]{(y+1)^4}\) can be written as \( (y+1)^{4/8}\). This transformation is crucial because it often makes subsequent simplification steps clearer.
Once rewritten, the goal is to identify if the expression can be reduced, relying on common mathematical techniques like fraction simplification or recognizing square or cube roots.

Fraction Simplification

Fraction simplification is a process where we reduce a fraction to its simplest form. In our example, we have the fraction \(\frac{4}{8}\), derived from the radical expression converted into a rational exponent form. Simplifying fractions involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
To simplify \(\frac{4}{8}\), notice that both 4 and 8 are divisible by 4, their GCD. So, dividing them each by 4 simplifies the fraction to \(\frac{1}{2}\). This key step helps in reducing the complexity of the original radical to a form that is much easier to work with.

• Simplified fractions make expressions more manageable.
• Non-simplified fractions could lead to more complex mathematics later on.

Understanding this process is essential when manipulating expressions involving rational exponents.

Properties of Exponents

The properties of exponents provide us with the rules that simplify expressions involving them. These properties simplify many mathematical expressions, especially those involving roots. In the original exercise, understanding that \(\frac{1}{2}\) exponent corresponds to the square root is crucial.
This knowledge comes from the property that states \(a^{m/n}\) is \(\sqrt[n]{a^m}\). Here, \( (y+1)^{1/2} \) is simplified to \(\sqrt{y+1}\). This clearly illustrates how rational exponents can be used to simplify expressions effectively.
By grasping these properties, students can dissect and solve more complex expressions. Always remember:

• Fractional exponents can simplify root operations.
• Reducing exponents makes it easier to visualize and solve expressions.

Mastery in handling exponents and their properties provides the mathematical tools needed for both basic and advanced problem-solving.