Problem 60 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive and write your answer in radical notation. $$ \sqrt[4]{x^{3}} \cdot \sqrt{x} $$

Step-by-Step Solution

Answer

\(\sqrt[4]{x^5}\)

Step 1: Rewrite Each Radical using Fractional Exponents

First, we'll rewrite the expression using fractional exponents. For the fourth root of \(x^3\), we can express it as \((x^3)^{1/4}\). For the square root of \(x\), it can be expressed as \(x^{1/2}\). Thus, the expression becomes: \((x^3)^{1/4} \times x^{1/2}\).

Step 2: Apply the Power of a Power Rule

When raising a power to another power, multiply the exponents. So, \((x^3)^{1/4}\) becomes \(x^{3 \times 1/4} = x^{3/4}\).

Step 3: Multiply the Exponents of the Same Base

Add the exponents together since the bases are the same (both are x). Therefore, \(x^{3/4} \times x^{1/2}\) becomes \(x^{3/4 + 1/2}\).

Step 4: Convert \(1/2\) to a Common Denominator with \(3/4\)

The fraction \(1/2\) needs to be converted to have the same denominator as \(3/4\). Convert \(1/2\) to \(2/4\).

Step 5: Add the Exponents

Add the fractions: \(3/4 + 2/4 = 5/4\). This means the expression can now be written as \(x^{5/4}\).

Step 6: Convert Back to Radical Notation

To express \(x^{5/4}\) in radical notation, it is written as \(\sqrt[4]{x^5}\), where the denominator of the fraction (4) becomes the root, and the numerator (5) becomes the power inside the radical.

Key Concepts

Radical Notation

In mathematics, radical notation is a way to express roots of numbers and variables. It is denoted by a radical symbol (√), and the number inside is called the radicand. For example, the expression \(\sqrt{x}\) is the square root of \(x\). When there's a number outside the radical, like \(\sqrt[4]{x}\), it indicates the fourth root of \(x\). This is extremely useful for simplifying and solving various algebraic expressions.

• Square Root: \(\sqrt{x}\) – The root index is 2.
• Fourth Root: \(\sqrt[4]{x}\) – The root index is 4.

Now, here's where it connects to our exercise. We start with \(\sqrt[4]{x^3}\) and \(\sqrt{x}\). Each radical can be rewritten into a simpler or different form known as fractional exponents, which we will look at next.

Fractional Exponents

Fractional exponents provide a powerful way to work with roots in algebraic expressions by turning them into powers that are often easier to manipulate. To convert a root into a fractional exponent, the general rule is:

• The denominator of the fraction is the index of the root.
• The numerator is the power of the radicand within the root.

For example, the fourth root of \(x^3\) becomes \((x^3)^{1/4}\). Meanwhile, the square root of \(x\) transforms into \(x^{1/2}\). By expressing roots in this way, you leverage the power rules of exponents to more readily combine and simplify expressions, like in our original problem.Once you've written radicals as fractional exponents, it becomes straightforward to use the rules of exponents, like multiplying exponents when a power is raised to another power, which is what we do when simplifying our problem further. Each of these conversions simplifies the process of combining expressions under the same base.

Simplification Steps

The journey to simplify the given expression \(\sqrt[4]{x^3} \cdot \sqrt{x}\) involves a few systematic steps: first, translating from radical notation to fractional exponents, and then solving.
1. **Convert to Fractional Exponents:**
- Change \(\sqrt[4]{x^3}\) to \((x^3)^{1/4} = x^{3/4}\).
- Turn \(\sqrt{x}\) into \(x^{1/2}\).
2. **Add Exponents:**
Since both parts have the same base \(x\), add the exponents: \(3/4\) from the fourth root and \(1/2\) from the square root.
- Convert \(1/2\) to \(2/4\) to have matching denominators.
- Adding gives \(x^{3/4 + 2/4} = x^{5/4}\).3. **Convert Back to Radical Notation:**
Express \(x^{5/4}\) back into radical form as \(\sqrt[4]{x^5}\).
By understanding each step, from conversion to adding exponents, you grasp how these components work together in algebra to simplify complex problems.