Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(\frac{-y^{3 / 2}}{y^{-1 / 3}}\right)^{3} $$

Step-by-Step Solution

Answer

-y^{11/2}

Step 1: Simplify the Fraction

We start by simplifying the fraction inside the parenthesis: \( \frac{-y^{3/2}}{y^{-1/3}} \). When dividing powers with the same base, we subtract exponents, so we get: \(-y^{(3/2) - (-1/3)} = -y^{3/2 + 1/3}\).

Step 2: Calculate Exponent Addition

To simplify the exponent further, we add the fractions \( \frac{3}{2} + \frac{1}{3} \). To do this, find a common denominator. The common denominator of 2 and 3 is 6, so we have: \( \frac{9}{6} + \frac{2}{6} = \frac{11}{6} \). Thus, the expression becomes \(-y^{11/6}\).

Step 3: Apply the Power to a Power Rule

Now, we apply the power of 3 to the entire expression \((-y^{11/6})^3\). According to the power of a power rule, \((a^m)^n = a^{m \cdot n}\), so we have \((-1)^3 \cdot y^{(11/6) \cdot 3}\).

Step 4: Simplify Each Part

First compute \((-1)^3\), which is \(-1\). Next, calculate \(y^{(11/6) \cdot 3}\). Simplifying the exponent gives us \(y^{33/6} = y^{11/2}\).

Step 5: Combine Results

Combine the results from Step 4 to get the final answer: \(-y^{11/2}\).

Key Concepts

Power of a Power Rule

When dealing with exponents, the power of a power rule is a key tool. It guides us on how to handle expressions where a power is raised to yet another power. The rule is simple: if you have

• expression: \((a^m)^n\)
• Then, it simplifies to:\(a^{m \cdot n}\)

This means you multiply the exponents together. In our example, we have \((-y^{11/6})^3\). The base \(-y\) is raised first to \(11/6\), then raised again to \(3\). Following the rule, we multiply \(11/6\) by \(3\), resulting in \((11/6) \cdot 3 = 33/6\), which simplifies to \(11/2\). So, the expression becomes \(-y^{11/2}\).

Exponent Addition

Adding exponents is essential when dividing like bases. When dividing, you subtract the exponents of the denominator from the exponents of the numerator. For our exercise:

• You start with: \(\frac{-y^{3/2}}{y^{-1/3}}\)
• Subtract the exponents: \((3/2) - (-1/3)\)

Converting to addition, we get \(3/2 + 1/3\). The next task is to find a common denominator for these fractions. Here, 6 serves as the common denominator, leading to: \(\frac{9}{6} + \frac{2}{6} = \frac{11}{6}\). So, the exponent we are dealing with becomes \(11/6\).

Simplifying Fractions

To simplify expressions involving fractions, it's crucial to manipulate fractional exponents correctly. Fractions in exponents often appear in expressions when performing operations like division. In our example:

• We had to simplify:\(-y^{3/2 + 1/3}\)
• This became:\(-y^{11/6}\)

Here, the fraction \(11/6\) shapes the new exponent following simplification techniques. Be sure to:

• Find a common denominator for addition or subtraction of exponents.
• Simplify the resulting fraction if necessary.

This approach aids in making the expression cleaner and easier to handle in subsequent steps.

Negative Exponents

Negative exponents might seem tricky, but they are straightforward to handle. A negative exponent suggests the inverse of the base with the positive exponent. The rule for negative exponents states \(a^{-m} = 1/a^m\). In our case, the negative sign in front of our base, \(-y\), persists throughout operations. It implies: we first deal with the magnitude of the exponent independently from the negative sign.
In simplifying, although the presence of a negative exponent provides direction for inversion, our task focused on progressing with the exponent operations themselves. Ultimately, this guideline helps ensure proper simplification, while understanding its impact will enhance your command over exponential functions.