Problem 40 - Algebra 2

Question

Simplify each expression. $$ \frac{r^{\frac{2}{3}}}{r^{\frac{1}{6}}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( r^{\frac{1}{2}} \).

Step 1: Apply the Quotient Rule of Exponents

To simplify the expression \( \frac{r^{\frac{2}{3}}}{r^{\frac{1}{6}}} \), we need to apply the quotient rule of exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \) for any nonzero base \( a \). So, we subtract the exponent in the denominator from the exponent in the numerator: \( r^{\frac{2}{3} - \frac{1}{6}} \).

Step 2: Find a Common Denominator for the Exponents

The exponents are \( \frac{2}{3} \) and \( \frac{1}{6} \). To subtract them, we first need a common denominator for \( \frac{2}{3} \) and \( \frac{1}{6} \). The least common multiple of 3 and 6 is 6, so we convert \( \frac{2}{3} \) to \( \frac{4}{6} \).

Step 3: Subtract the Exponents

Now, we have \( r^{\frac{4}{6} - \frac{1}{6}} \). Subtract the exponents: \( \frac{4}{6} - \frac{1}{6} = \frac{3}{6} \).

Step 4: Simplify the Resulting Exponent

The exponent \( \frac{3}{6} \) can be simplified by dividing the numerator and the denominator by their greatest common divisor, 3. Thus, \( \frac{3}{6} = \frac{1}{2} \).

Step 5: Write the Final Expression

After simplification, the expression becomes \( r^{\frac{1}{2}} \).

Key Concepts

Quotient Rule of Exponents

One of the important exponent rules you need to know is the Quotient Rule of Exponents. This rule is incredibly helpful when you have the same base raised to different powers being divided. The rule states that

• \( \frac{a^m}{a^n} = a^{m-n} \)

This means you can simply subtract the exponent of the denominator from the exponent of the numerator.
It simplifies working with fractional exponents too, as seen in expressions like \( \frac{r^{\frac{2}{3}}}{r^{\frac{1}{6}}} \).
By applying this rule, you make calculations more straightforward and avoid unnecessary complexity.
It's always a good practice to check if the bases of the exponents are the same first. If they are, the Quotient Rule of Exponents can be directly applied.

Simplifying Expressions

Simplifying expressions, especially those with exponents, requires a step-by-step approach to ensure clarity and correctness. Begin by understanding what operations need to occur, specifically when using exponent rules.
When you apply the Quotient Rule of Exponents, sometimes you end up with fractional exponents, which need to be simplified further. For example, after applying the rule to \( \frac{r^{\frac{2}{3}}}{r^{\frac{1}{6}}} \), you get \( r^{\frac{2}{3} - \frac{1}{6}} \).
It’s important to simplify the resulting exponent, making it easier to interpret and use in subsequent calculations.
Always take care to break down the problem and resolve each part meticulously. This prevents errors and helps in fully understanding the nuances of exponentiation.

Common Denominator

Working with fractional exponents often requires you to find a common denominator. This is particularly true when you need to subtract or add exponents. The common denominator allows for a smooth transition into subtraction or addition of fractions.
Consider the exponents \( \frac{2}{3} \) and \( \frac{1}{6} \). Their least common denominator is 6, which transforms the fractions into equivalent forms with this common base.

• \( \frac{2}{3} \) becomes \( \frac{4}{6} \)


• \( \frac{1}{6} \) stays \( \frac{1}{6} \)

Now, you can perform subtraction: \( \frac{4}{6} - \frac{1}{6} = \frac{3}{6} \). Simplifying \( \frac{3}{6} \) further by dividing both the numerator and the denominator by 3 results in \( \frac{1}{2} \).
This makes it easier to not only perform calculations but also understand the relationships between fractional exponents.