Problem 65 - Algebra 2

Question

Simplify each expression. Assume that all variables are positive. $$x^{\frac{2}{7}} \cdot x^{\frac{3}{14}}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{\frac{1}{2}}\).

Step 1: Analyze the expression

The given expression is \(x^{\frac{2}{7}}\cdot x^{\frac{3}{14}}\). Here, both terms have the same base of x but different fractional exponent.

Step 2: Find common denominator

To add these fractions, need to find a common denominator. The least common multiple of 7 and 14 is 14. So, rewrite \(\frac{2}{7}\) as \(\frac{4}{14}\).

Step 3: Apply the Law of Exponents

According to the laws of exponents, when bases are the same, we add the exponents. So, add \(\frac{4}{14}\) and \(\frac{3}{14}\) to get \(\frac{7}{14}\).

Step 4: Simplify the expression

Simplify the exponent \(\frac{7}{14}\) to \(\frac{1}{2}\). So, the simplified expression is \(x^{\frac{1}{2}}\).

Key Concepts

Laws of Exponents

Understanding the laws of exponents is crucial when dealing with fractional exponents. These laws help in simplifying expressions that might initially seem complicated. One key principle is that when you multiply expressions that have the same base, you add their exponents. This is particularly handy when you encounter expressions such as \( x^{\frac{a}{b}} \cdot x^{\frac{c}{d}} \). By adding the exponents, the expression becomes easier to manage.

Here, both terms share the same base \( x \) with different fractional exponents. Thus, by applying the law of exponents, we simplify \( x^{\frac{2}{7}} \cdot x^{\frac{3}{14}} \) by adding \( \frac{2}{7} \) and \( \frac{3}{14} \). Once the fractions have a common denominator, this becomes straightforward.

Simplification

The simplification process is where you transform a complex mathematical expression into a simpler form. It's important in making calculations manageable and also in understanding the underlying mathematical relationships in an expression.

In the case of \( x^{\frac{2}{7}} \cdot x^{\frac{3}{14}} \), simplification begins by finding a common denominator to ease the addition of fractional exponents. After achieving this, the next step is to add the fractions: \( \frac{4}{14} + \frac{3}{14} = \frac{7}{14} \). Simplification continues by reducing the fraction \( \frac{7}{14} \) to its simplest form \( \frac{1}{2} \).

This results in a more straightforward expression \( x^{\frac{1}{2}} \), which is much easier to comprehend and work with. In general, simplification is about making expressions as neat as possible.

Common Denominator

A common denominator is essential when you are adding or subtracting fractions. It is a shared multiple of the denominators of the fractions involved. Finding a common denominator allows us to easily combine fractions.

In mathematical expressions involving exponents like \( x^{\frac{2}{7}} \) and \( x^{\frac{3}{14}} \), determining the common denominator enhances the process of addition of the exponents. In this particular example, the denominators \( 7 \) and \( 14 \) need a common base to be added effectively.

• The least common multiple of both numbers \( 7 \) and \( 14 \) is \( 14 \).
• This allows us to rewrite \( \frac{2}{7} \) as \( \frac{4}{14} \).

Once both fractions share the denominator \( 14 \), their addition \( \frac{4}{14} + \frac{3}{14} \) becomes a simple matter. Obtaining a common denominator simplifies operations involving fractions, leading to a simplified answer.