Problem 574 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}\) (b) \(\frac{y^{\frac{5}{2}}}{y^{\frac{1}{2}}}\) ( \() \frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}\)

Step-by-Step Solution

Answer

a) x, b) y^2, c)\frac{1}{r}.

Step 1: Simplify \(\frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}\)

To simplify the fraction \(\frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}\), subtract the exponent of the denominator from the exponent of the numerator. This is based on the property \(\frac{a^m}{a^n} = a^{m-n}\). So, we have: \[ \frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}} = x^{\frac{7}{2} - \frac{5}{2}} = x^{\frac{2}{2}} = x^1 = x \].

Step 2: Simplify \(\frac{y^{\frac{5}{2}}}{y^{\frac{1}{2}}}\)

Using the same property \(\frac{a^m}{a^n} = a^{m-n}\), subtract the exponents: \[ \frac{y^{\frac{5}{2}}}{y^{\frac{1}{2}}} = y^{\frac{5}{2} - \frac{1}{2}} = y^{\frac{4}{2}} = y^2 \].

Step 3: Simplify \(\frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}\)

Again, using \(\frac{a^m}{a^n} = a^{m-n}\), subtract the exponents: \[ \frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}} = r^{\frac{4}{5} - \frac{9}{5}} = r^{\frac{-5}{5}} = r^{-1} = \frac{1}{r} \].

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential for simplifying expressions involving powers. One key property is the quotient rule, which states that when dividing like bases with different exponents, you subtract the exponent of the denominator from the exponent of the numerator. This can be formulated as \(\frac{a^m}{a^n} = a^{m-n}\).
This property helps in transforming complex expressions into simpler ones. For instance, when dealing with fractional exponents, the rule still applies. This allows us to handle bases raised to different fractional powers effectively.

Fractional Exponents

Fractional exponents may seem intimidating at first, but they are just another way to represent roots. For example, \(x^{\frac{1}{2}}\) is equivalent to \sqrt{x}\, and \(x^{\frac{1}{3}}\) represents the cube root of \(x\).
When working with fractional exponents in division, the same exponent rules apply. The exponents are manipulated using the properties of exponents, making calculations more streamlined. For example, simplifying \( \frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}\) involves subtracting the exponents: \(\frac{7}{2} - \frac{5}{2} = 1\), which simplifies to \x\.

Step-by-Step Solutions

Breaking down problems into simple steps makes them easier to solve and understand. Let's apply this to the original exercise.

**Step 1:** Simplify \(\frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}\). Use the quotient rule: \(\frac{7}{2} - \frac{5}{2} = \frac{2}{2} = 1\), so the result is \(x\).

**Step 2:** Simplify \(\frac{y^{\frac{5}{2}}}{y^{\frac{1}{2}}}\). Again, use the quotient rule: \(\frac{5}{2} - \frac{1}{2} = \frac{4}{2} = 2\), so the result is \(y^2\).

**Step 3:** Simplify \(\frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}\). Using the same property: \(\frac{4}{5} - \frac{9}{5} = \frac{-5}{5} = -1\), this gives \(r^{-1} = \frac{1}{r}\).

Regardless of complexity, following a step-by-step approach ensures clarity and accuracy. This method shields against errors and builds confidence in problem-solving.