Problem 25 - College Algebra with Modeling and Visualization

Question

Use positive exponents to rewrite. $$ \sqrt{x} \cdot \sqrt[3]{x} $$

Step-by-Step Solution

Answer

The expression simplifies to \(x^{5/6}\).

Step 1: Expressing Square Root with Exponents

Notice that the square root of a number can be expressed as a fractional exponent. The square root of \(x\) is the same as \(x^{1/2}\). So, replace \(\sqrt{x}\) with \(x^{1/2}\).

Step 2: Expressing Cube Root with Exponents

Similarly, the cube root of a number can be expressed as a fractional exponent as well. The cube root of \(x\) is the same as \(x^{1/3}\). So, replace \(\sqrt[3]{x}\) with \(x^{1/3}\).

Step 3: Multiplying with Exponents

When multiplying expressions with the same base, you add the exponents. According to the rule of exponents \(a^m \cdot a^n = a^{m+n}\), you have \(x^{1/2} \cdot x^{1/3} = x^{(1/2 + 1/3)}\).

Step 4: Adding the Exponents

To add the exponents \(1/2\) and \(1/3\), find a common denominator. The least common denominator of 2 and 3 is 6. Convert \(1/2\) to \(3/6\) and \(1/3\) to \(2/6\). Thus, \(\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Step 5: Writing in Powers

With the exponents added, the expression \(x^{1/2} \cdot x^{1/3}\) simplifies to \(x^{5/6}\).

Key Concepts

Fractional Exponents

Fractional exponents are an intriguing aspect of mathematics. They allow us to express roots in the language of powers. Anytime you see a root symbol, like a square root or cube root, you can convert it into a fractional exponent for simplification.
For example, the square root of a number, such as \( \sqrt{x} \), is equivalent to \( x^{1/2} \). Similarly, the cube root of \( x \), shown as \( \sqrt[3]{x} \), is represented as \( x^{1/3} \).
By using fractional exponents, you can make complex expressions more manageable, especially when combined with other operations such as multiplication or division.

• Square root \( \sqrt{x} = x^{1/2} \)
• Cube root \( \sqrt[3]{x} = x^{1/3} \)
• Fourth root \( \sqrt[4]{x} = x^{1/4} \) and so forth.

By practicing with fractional exponents, it becomes easier to solve various algebraic expressions.

Roots and Radicals

Roots and radicals are all about finding which number, when multiplied by itself a specific number of times, gives the original number. The square root is the most common radical, but there are many other types.
When you see \( \sqrt{x} \), it's telling you to find a number that multiplies by itself to yield \( x \). This is written mathematically as \( x^{1/2} \). For cube roots, such as \( \sqrt[3]{x} \), you're finding which number cubed equals \( x \), and this is written as \( x^{1/3} \).
Familiarity with roots and radicals is very useful in solving equations and simplifying expressions.

• Helps in solving equations where variables are under a root.
• Can simplify calculations by transforming them into smaller parts.

Learning about roots and radicals not only enhances your math toolbox but also supports your understanding of fundamental operations.

Multiplication of Exponents

The multiplication of exponents is based on a simple yet powerful rule. When you multiply terms with the same base, you add the exponents.
Consider expressions like \( a^m \cdot a^n \). The multiplication rule tells us that this equals \( a^{m+n} \). Essentially, you keep the base the same and just add up the exponents. This is exactly what happens when you multiply \( x^{1/2} \cdot x^{1/3} \), resulting in \( x^{(1/2 + 1/3)} \).
To perform this operation, you may need a common denominator to add the exponents:

• Convert \( 1/2 \) to \( 3/6 \)
• Convert \( 1/3 \) to \( 2/6 \)
• Add: \( 3/6 + 2/6 = 5/6 \)

So the entire expression simplifies nicely to \( x^{5/6} \), following the exponent rules.