Problem 93 - Elementary and Intermediate Algebra

Question

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[4]{5^{2}} $$

Step-by-Step Solution

Answer

\(\sqrt[4]{5^2} = \sqrt{5}\)

Step 1: Express the Radical Using Rational Exponents

The expression involves a fourth root, which can be represented using rational exponents. Recall that the n-th root of a number can be written as a rational exponent:\[\sqrt[n]{x} = x^{1/n}\]Therefore, the fourth root of \(5^2\) is:\[(5^2)^{1/4}\]

Step 2: Multiply the Exponents

Apply the property of exponents, which states that \((a^m)^n = a^{m \cdot n}\). So, multiply the exponents inside the parentheses:\[(5^2)^{1/4} = 5^{2 \cdot (1/4)} = 5^{2/4}\]

Step 3: Simplify the Fraction

The fraction \(\frac{2}{4}\) simplifies to \(\frac{1}{2}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:\[5^{2/4} = 5^{1/2}\]

Step 4: Express the Result in Simplified Form

The expression with the simplified exponent \(\frac{1}{2}\) represents a square root. Therefore, \(5^{1/2}\) is equivalent to \(\sqrt{5}\):\[5^{1/2} = \sqrt{5}\]

Key Concepts

Simplifying Radicals

Radicals can often seem daunting at first, with their intimidating root symbols like square roots (\(\sqrt{}\)) and cube roots (\(\sqrt[3]{}\)). However, transforming these into rational exponents can make them much easier to handle. The n-th root of any number \(x\) can be expressed as a rational exponent, converting the expression \(\sqrt[n]{x}\) into \(x^{1/n}\). This transformation is incredibly powerful because it allows us to use exponent rules to simplify complex expressions easily.

• Start by identifying the type of root in the expression. For example, a fourth root becomes an exponent of \(1/4\).
• Rewrite the entire expression with this new exponent. This step turns the radical into a more manageable form.

Understanding how to manipulate expressions with rational exponents will give you the flexibility to simplify or transform these expressions further.

Exponent Rules

Exponent rules are crucial for simplifying and manipulating expressions involving powers and roots. Let’s dive into a few key rules that come in handy often:

• Product of Powers Rule: Multiply the exponents when raising a power to another power, like \((a^m)^n = a^{m\cdot n}\).
• Quotient of Powers Rule: When dividing similar bases, subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Product Rule: Distribute the exponent to each part of the product: \((ab)^n = a^n b^n\).
• Negative Exponent Rule: Flip the base to turn a negative exponent into a positive one: \(a^{-n} = 1/a^n\).

By applying these rules, you can tackle complex expressions with confidence. For instance, when simplifying \((5^2)^{1/4}\), you multiply the exponents inside to get \(5^{2/4}\), streamlining the simplification process efficiently.

Fraction Simplification

Simplifying fractions forms the backbone of manipulating expressions involving rational exponents. As you already know, fractions consist of two numbers: the numerator on top and the denominator on the bottom. The aim is to express fractions in their simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).

• Identify the GCD of the numerator and the denominator. For \(\frac{2}{4}\), the GCD is 2.
• Divide both parts of the fraction by the GCD to simplify: \(\frac{2}{4} \Rightarrow \frac{2 \div 2}{4 \div 2} = \frac{1}{2}\).

When dealing with decimal fractions, reducing them to their simplest terms makes further calculations more straightforward. This process is essential in simplifying expressions with rational exponents to create clear, concise mathematical solutions.