Problem 10 - Algebra 2

Question

Simplify each expression. $$ a^{\frac{2}{3}} \cdot a^{\frac{1}{4}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( a^{\frac{11}{12}} \).

Step 1: Apply the Product Rule of Exponents

The product rule of exponents states that when you multiply two exponents with the same base, you add the exponents. Here, the base is \( a \) and the exponents are \( \frac{2}{3} \) and \( \frac{1}{4} \). So, we have the expression: \[ a^{\frac{2}{3}} \cdot a^{\frac{1}{4}} = a^{\left(\frac{2}{3} + \frac{1}{4}\right)}.\]

Step 2: Find a Common Denominator

To add the exponents \( \frac{2}{3} \) and \( \frac{1}{4} \), first find a common denominator. The least common denominator of 3 and 4 is 12. Convert \( \frac{2}{3} \) to \( \frac{8}{12} \) and \( \frac{1}{4} \) to \( \frac{3}{12} \).

Step 3: Add the Exponents

Now add the exponents with the common denominator:\[ \frac{8}{12} + \frac{3}{12} = \frac{11}{12}. \] So, the expression becomes: \[ a^{\frac{11}{12}}. \]

Step 4: Simplified Expression

The simplified form of the original expression \( a^{\frac{2}{3}} \cdot a^{\frac{1}{4}} \) is: \[ a^{\frac{11}{12}}. \] This is the simplest form we can achieve through exponent rules.

Key Concepts

Simplifying Expressions

Simplifying expressions involves making them as straightforward as possible without changing their value. In mathematics, this is a crucial skill that makes complex problems easier to solve. It often involves performing operations like addition, subtraction, multiplication, and division on variables and numbers within an equation.
To simplify an expression involving exponents, it's important to utilize exponent laws such as the product rule. By doing so, we can transform combinations of exponents into equivalent simpler forms.
For instance, in some cases, you might need to combine like terms, or reduce fractions in the expression. Think of simplifying as cleaning up the expression to its neatest form.

Exponential Expressions

Exponential expressions are a form of notation that allows us to represent repeated multiplication in a compact way. They include a base, which is the number being multiplied, and an exponent, which tells us how many times the base is multiplied by itself.
The product rule of exponents is particularly handy when working with exponential expressions of the same base. It states: when you multiply exponents with the same base, you can add their exponents. For example, multiplying \( a^m \cdot a^n \) results in \( a^{m+n} \).
This rule simplifies the process of working with exponential expressions by transforming a problem involving multiplication into a simpler property of addition.

Fractions and Common Denominators

Fractions are parts of a whole and are crucial in numerous areas of mathematics, including when dealing with exponential expressions. When adding or subtracting fractions, a common denominator is required.
Finding the least common denominator (LCD) means identifying the smallest multiple that two or more denominators share. This allows fractions to be easily added or subtracted since it converts them to equivalent fractions with the same denominator.
For example, when you have the fractions \( \frac{2}{3} \) and \( \frac{1}{4} \), the least common denominator is 12. Converting each fraction gives \( \frac{8}{12} \) and \( \frac{3}{12} \). Adding these becomes straightforward, leading to the combined fraction \( \frac{11}{12} \). This method is invaluable in simplifying exponential expressions with fractional exponents, as seen when using the product rule.