Problem 39 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[4]{\frac{3}{625}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{\sqrt[4]{3}}{5} \).

Step 1: Express the Radical Expression Using Exponents

Rewrite the fourth root as an expression with a fractional exponent. The expression \( \sqrt[4]{\frac{3}{625}} \) can be rewritten using exponents as \( \left( \frac{3}{625} \right)^{\frac{1}{4}} \).

Step 2: Simplify the Denominator

Next, recognize that \( 625 \) can be rewritten as \( 5^4 \) because \( 5 \times 5 \times 5 \times 5 = 625 \). Therefore, \( \frac{3}{625} \) can be rewritten as \( \frac{3}{5^4} \).

Step 3: Apply the Exponent to the Denominator

Apply the exponent \( \frac{1}{4} \) to the term \( \frac{3}{5^4} \). This results in \( \left( \frac{3}{5^4} \right)^{\frac{1}{4}} = \frac{3^{\frac{1}{4}}}{(5^4)^{\frac{1}{4}}} \).

Step 4: Simplify the Denominator Further

Simplify \((5^4)^{\frac{1}{4}}\) by recognizing that when you raise a power to a power, you multiply the exponents: \((5^4)^{\frac{1}{4}} = 5^{\frac{4}{4}} = 5^1 = 5\).

Step 5: Combine to Simplify the Entire Expression

Now rewrite the expression \( \frac{3^{\frac{1}{4}}}{5} \) in radical form as \( \frac{\sqrt[4]{3}}{5} \). Thus, the entire expression \( \sqrt[4]{\frac{3}{625}} \) simplifies to \( \frac{\sqrt[4]{3}}{5} \).

Key Concepts

Fractional Exponents

Fractional exponents provide a more general way of expressing roots and powers in mathematics. They offer an alternative to the radical notation, making algebraic operations easier to manage. When we talk about fractional exponents, we're essentially describing roots. For example, the fourth root of a number can be written with an exponent of \( \frac{1}{4} \). This means any number raised to the power of \( \frac{1}{4} \) is equivalent to taking its fourth root.

• The numerator of the exponent (if not 1) indicates the power to which the base is raised.
• The denominator of the exponent tells you the root you are taking. For instance, \( a^{\frac{m}{n}} \) signifies the \(n\)-th root of \(a^m\).

Converting radical expressions into terms with fractional exponents can simplify complex calculations, especially when multiplying or dividing exponents with different bases.

Simplifying Exponents

Simplifying exponents involves reducing expressions to their most manageable form, often making calculations easier. A crucial rule to remember is the power of a power rule, which states \((a^m)^n = a^{m \times n}\). This rule helps break down expressions with multiple exponents, making them simpler to solve. Consider this example from the original exercise:

• For the denominator in \( \frac{3}{625} \), which is \(5^4\), taking the fourth root applied as a fractional exponent, \( \left(5^4\right)^{\frac{1}{4}}\), simplifies by multiplying exponents to give \(5^{\frac{4}{4}} = 5^1 = 5\).

By simplifying the terms step by step, you can systematically navigate from complex expressions down to their simplest forms.

Fourth Roots

Understanding fourth roots can deepen your grasp of radical expressions. A fourth root is a special type of root that asks the question: "What number, when multiplied by itself four times, gives the original number?"To find a fourth root, we express it with a fractional exponent of \( \frac{1}{4} \). Let’s look at a breakdown of the fourth root of a base, like in our original exercise:

• Starting with the expression \( \sqrt[4]{x} \), it can be rewritten as \( x^{\frac{1}{4}} \).
• If you know the expression contains a base that is a perfect fourth power, such as \( 16 \) or in our exercise's case with \( 625 = 5^4 \), then the fourth root simplifies directly to the base, like \(5\) here.

Working through problems involving fourth roots involves recognizing these bases and applying fractional exponents to clearly understand and simplify the expression. Understanding fourth roots is essential in tackling more complex algebraic problems efficiently.