Problem 48 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{x^{3 / 4}}{x^{1 / 8}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^{5/8} \).

Step 1: Understanding the Problem

The problem asks us to simplify the expression \( \frac{x^{3/4}}{x^{1/8}} \) using the properties of exponents, ensuring that all exponents in the final result are positive.

Step 2: Applying the Quotient of Powers Property

The Quotient of Powers Property states that \( \frac{x^a}{x^b} = x^{a-b} \). We'll apply this property to the expression \( \frac{x^{3/4}}{x^{1/8}} \), giving us \( x^{3/4 - 1/8} \).

Step 3: Finding a Common Denominator

To subtract the exponents, we need a common denominator. For \( \frac{3}{4} \) and \( \frac{1}{8} \), the least common denominator is 8. Thus, \( \frac{3}{4} = \frac{6}{8} \).

Step 4: Subtracting the Exponents

Now, subtract \( \frac{1}{8} \) from \( \frac{6}{8} \): \[ \frac{6}{8} - \frac{1}{8} = \frac{5}{8}. \] So the expression becomes \( x^{5/8} \).

Step 5: Conclusion

The simplified expression is \( x^{5/8} \), where the exponent is positive.

Key Concepts

Quotient of Powers

The quotient of powers is a fundamental concept in understanding how to simplify expressions with exponents. When you see an expression like \( \frac{x^a}{x^b} \), the quotient of powers property helps us by allowing us to subtract the exponents: \( x^{a-b} \). This is valid because when dividing like bases, the exponents indicate how many times the base gets multiplied.

For example, in the expression \( \frac{x^{3/4}}{x^{1/8}} \), we can write it as \( x^{3/4 - 1/8} \). By subtracting the exponent of the denominator from the numerator, we effectively reduce the expression to a simpler form, which is crucial in math for finding solutions quickly and accurately.

Remembering this property will save you time and help avoid confusion when dealing with more complex expressions.

Simplification

Simplification is a process of making expressions easier to work with, by reducing them to the simplest form. In the context of exponents, simplification often involves eliminating fractional or negative exponents, and expressing the result in terms of positive ones.

The key to effective simplification is understanding the relationships between the components of your expression. For instance, when you're given \( \frac{x^{3/4}}{x^{1/8}} \), simplifying it requires first applying the quotient of powers property to combine the exponents.

Then, you need to ensure you're working with like terms, which often means finding a common denominator. For the fractions \( \frac{3}{4} \) and \( \frac{1}{8} \), this means converting both to eighths, so you can easily subtract them.

Finally, the expression \( x^{5/8} \) is considered simplified because it has been reduced to its simplest form with positive exponents.

Positive Exponents

Using positive exponents is often a desired result when simplifying expressions, because it provides clarity and avoids potential errors related to negative exponents. Positive exponents are straightforward, indicating how many times a base should be multiplied by itself.

In some exercises, like the one previously mentioned, the directions specifically ask for positive exponents. If your expressions result in negative exponents, remember you can convert them using reciprocal functions; however, be vigilant when the goal is to keep all exponents positive from the start.

In our example where we ended up with \( x^{5/8} \), this is already in a positive exponent form, which makes it easier to interpret and apply in further calculations. Keeping exponents positive not only simplifies the expression but is also a helpful practice when verifying your work.