Problem 79 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \( x^{1/2} y^{1/2} \).

Step 1: Rewrite the Radical Using Rational Exponents

The expression \( \sqrt[8]{x^{4} y^{4}} \) involves an eighth root. We can rewrite this radical expression using rational exponents. Recall that \( \sqrt[n]{a} = a^{1/n} \). Therefore, \( \sqrt[8]{x^4 y^4} \) becomes \( (x^4 y^4)^{1/8} \).

Step 2: Apply the Power of a Product Rule

Using the power of a product rule, \( (ab)^m = a^m b^m \), we can further simplify the expression \( (x^4 y^4)^{1/8} \) as \( (x^4)^{1/8} (y^4)^{1/8} \).

Step 3: Simplify Each Power Separately

Simplify each part separately by applying the property \( (a^m)^n = a^{mn} \). Thus, \( (x^4)^{1/8} = x^{4/8} = x^{1/2} \) and \( (y^4)^{1/8} = y^{4/8} = y^{1/2} \).

Step 4: Combine the Simplified Expressions

Combine the simplified expressions from Step 3 to get the final answer. Thus, the simplified form of \( \sqrt[8]{x^{4} y^{4}} \) is \( x^{1/2} y^{1/2} \).

Key Concepts

Understanding Radicals

Radicals are expressions that involve roots. You often encounter square roots, like \( \sqrt{a} \), but there are other types like cube roots \( \sqrt[3]{a} \), and in this case, eighth roots, \( \sqrt[8]{a} \). The root simply indicates that you want to find the number which, when raised to a certain power (equal to the root), gives back the original number under the radical. For square roots, this number is 2, for cube roots it's 3, and for eighth roots, it’s 8.
To handle the particular radical \( \sqrt[8]{x^4 y^4} \), we rewrite it using rational exponents. Each radical can be expressed as a fractional exponent. Thus, \( \sqrt[8]{x^4 y^4} \) is the same as \( (x^4 y^4)^{1/8} \). Rational exponents offer a simplified way to deal with roots, using the rule \( \sqrt[n]{a} = a^{1/n} \).
This transformation also sets up further simplification and manipulation possible through exponent rules.

Simplifying Expressions with Rational Exponents

Simplifying expressions involves reducing them to their simplest form. When we are working with rational exponents, we apply specific rules that streamline the process. One of these is the 'Power of a Product Rule' which states \( (ab)^m = a^m b^m \). This allows us to break down more complex expressions into simpler components.
In the expression \((x^4 y^4)^{1/8}\), applying this rule lets us separate it into \((x^4)^{1/8} (y^4)^{1/8}\). This split means we can tackle each component individually, focusing on \(x\) and \(y\) separately.
Once split, you ensure each component is in its simplest form by applying the rule \((a^m)^n = a^{mn}\). Thus, \((x^4)^{1/8}\) simplifies to \(x^{4/8} = x^{1/2}\), while \((y^4)^{1/8}\) becomes \(y^{4/8} = y^{1/2}\). The result is a cleaner and much simpler expression.

The Importance of Exponent Rules

Exponent rules simplify the way we deal with powers and roots in mathematics. Knowing these rules provides an efficient way to transform and reduce expressions, particularly those involving radicals and rational exponents.
Here is a quick recap of some essential rules:

• The 'Product Rule' (\(a^m a^n = a^{m+n}\)) combines powers with the same base by adding their exponents.
• The 'Power of a Product Rule' (\((ab)^m = a^m b^m\)) allows for separating products into smaller, more manageable parts.
• The 'Power of a Power Rule' (\((a^m)^n = a^{mn}\)) is especially useful for breaking down complex exponent expressions into simpler terms.

By applying these exponent rules, expressions like \( \sqrt[8]{x^4 y^4} \) become straightforward when rewritten as \(x^{1/2} y^{1/2}\).
Understanding and utilizing these rules not only simplifies your current calculations but is also instrumental in solving more advanced mathematical problems.