Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\frac{\sqrt[3]{x}}{\sqrt{x}}$$

Step-by-Step Solution

Answer

\(x^{-\frac{1}{6}}\)

Step 1: Rewrite Expressions as Exponents

The cube root of a number \( x \) can be expressed as \( x^{\frac{1}{3}} \), and the square root of \( x \) can be expressed as \( x^{\frac{1}{2}} \). The given expression becomes: \[ \frac{x^{\frac{1}{3}}}{x^{\frac{1}{2}}} \]

Step 2: Subtract Exponents

When dividing two expressions with the same base, subtract their exponents: \( x^a / x^b = x^{a-b} \). Therefore, \[ \frac{x^{\frac{1}{3}}}{x^{\frac{1}{2}}} = x^{\frac{1}{3} - \frac{1}{2}} \]

Step 3: Simplify the Exponent

Calculate \( \frac{1}{3} - \frac{1}{2} \). To do this, find a common denominator, which is 6, and rewrite the fractions: \( \frac{1}{3} = \frac{2}{6} \) and \( \frac{1}{2} = \frac{3}{6} \). Thus, \( \frac{2}{6} - \frac{3}{6} = -\frac{1}{6} \).

Step 4: Rewrite the Expression

Substitute the simplified exponent back into the expression: \[ x^{-\frac{1}{6}} \]

Key Concepts

Positive Numbers in Rational Exponents

When working with exponents, especially rational ones, it's important to remember that we often deal with positive numbers. This is because the rules of exponents become simpler and more predictable. For instance, when you have a positive base, any exponent will follow standard multiplication rules without involving complex numbers.

• A rational exponent means we're using fractions as exponents, like \( x^{\frac{1}{3}} \), which can also mean taking roots of numbers.
• In this exercise, assuming variables represent positive numbers allows us to use these standard exponent rules comfortably and gives us predictable positive outcomes.
• Roots and fractional powers, due to their nature, are more straightforward to compute with positive numbers.

Understanding this foundational concept helps make simplifying expressions with rational exponents such as square roots or cube roots a neat and efficient process.

Exponent Subtraction

Exponent subtraction is a vital skill when simplifying expressions where variables with exponents are divided. The rule is clear: when dividing like bases, subtract the exponent in the denominator from the one in the numerator.
Imagine we have \( \frac{x^a}{x^b} \). This simplifies to \( x^{a-b} \). So long as the variable \( x \) does not equal zero, this rule works flawlessly.

• In our exercise, \( x^{\frac{1}{3}} \) divided by \( x^{\frac{1}{2}} \) uses this exact rule.
• The result is \( x^{\frac{1}{3} - \frac{1}{2}} \), neatly simplifying to a single exponent term.
• Such operations emphasize the importance of knowing how rational exponents work in division.

This makes it easier to manipulate and simplify expressions in algebra.

Finding a Common Denominator for Exponent Subtraction

One hurdle when subtracting rational exponents is finding a common denominator. This step is crucial because it allows us to perform arithmetic on fractions easily. By expressing fractions with a common denominator, subtraction becomes a straightforward task.

Let's look at our example of \( \frac{1}{3} - \frac{1}{2} \). To subtract these, we need a common denominator:

• The smallest common multiple of 2 and 3 is 6.
• Rewriting both fractions with this denominator gives us \( \frac{1}{3} = \frac{2}{6} \) and \( \frac{1}{2} = \frac{3}{6} \).
• Then, \( \frac{2}{6} - \frac{3}{6} = -\frac{1}{6} \).

With a common denominator in place, we accurately find the difference, paving the way to simplify the original expression into something more manageable, \( x^{-\frac{1}{6}} \). This illustrates why finding a common denominator is an integral part of working with rational exponents.