Problem 42 - Precalculus: Functions and Graphs

Question

Simplify. \(\left(\frac{-y^{3 / 2}}{y^{-1 / 3}}\right)^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \(-y^{11/2}\).

Step 1: Apply the Quotient Rule of Exponents

The expression to simplify is \(\left(\frac{-y^{3/2}}{y^{-1/3}}\right)^3\). According to the quotient rule for exponents, \( \frac{a^m}{a^n} = a^{m-n} \). Apply this rule to simplify \( \frac{-y^{3/2}}{y^{-1/3}} \):\[ -y^{3/2 - (-1/3)} = -y^{3/2 + 1/3} \]

Step 2: Simplify the Exponents

Now calculate the addition in the exponent from the previous step:\[ 3/2 + 1/3 = \frac{9}{6} + \frac{2}{6} = \frac{11}{6} \]Thus, the expression becomes:\[ -y^{11/6} \]

Step 3: Apply the Power of a Power Rule

Now apply the power of a power rule which states \((a^m)^n = a^{m \cdot n}\). Use this rule to simplify \((-y^{11/6})^3\):\[ (-y^{11/6})^3 = (-1)^3 \cdot y^{(11/6) \cdot 3} = -y^{11/2} \]

Step 4: Calculate the Final Exponent Product

Complete the calculation for the exponent:\[(11/6) \cdot 3 = \frac{33}{6} = \frac{11}{2}\]Thus, the simplified expression is:\[-y^{11/2}\]

Key Concepts

Quotient Rule of Exponents

The Quotient Rule of Exponents is a helpful tool when simplifying expressions involving fractions with like bases. When you divide exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
This can be remembered with the rule: \( \frac{a^m}{a^n} = a^{m-n} \). For example, if you have \( \frac{y^{3/2}}{y^{-1/3}} \), you subtract the exponents: \( 3/2 - (-1/3) \).

• First, consider the sign of each exponent. Negative exponents mean division, which is the opposite of multiplication.
• Next, change the subtraction of the negative exponent to an addition: \( 3/2 + 1/3 \).
• Lastly, simplify the fraction: \( 3/2 + 1/3 \) becomes \( \frac{11}{6} \).

Understanding this rule is key for managing and simplifying complex expressions with ease.

Power of a Power Rule

The Power of a Power Rule explains how to simplify expressions where an exponent is raised to another power. This is done by multiplying the exponents together, as shown by the rule \((a^m)^n = a^{m \cdot n}\).
In the exercise, once we simplified the quotient, we had the expression \((-y^{11/6})^3\). Here, we'll apply the power rule.

• Look at the expression: each base and its exponent, like \(y^{11/6}\), is raised to an additional power, \(3\).
• Multiply the exponents for \(y\): \((11/6) \cdot 3\) which simplifies to \(\frac{33}{6} = \frac{11}{2}\).
• Do not forget to apply the power to any numerical values as well, for instance, \((-1)^3 = -1\).

The result becomes \(-y^{11/2}\), combining both the influence of the base and the processing through exponent rules.

Rational Exponents

Rational exponents might seem tricky at first, but they're simply another way to express roots and powers. A rational exponent like \(a^{p/q}\) is another way of writing the \(q\)-th root of \(a^p\).
It's important to recognize that the numerator of a rational exponent represents a power, while the denominator represents a root, like \(a^{3/2} = (a^3)^{1/2}\).
Here are some helpful points to grasp the concept:

• Translation: \(a^{m/n} = \sqrt[n]{a^m}\), offering a direct path between roots and exponents.
• Simplifying: Rational exponents can often be simpler to handle than radical signs, especially in algebraic expressions.
• Transformation: You can convert unusual roots into rational exponents, making use of exponentiation rules to simplify calculations.

Understanding rational exponents is key in algebra as they frequently appear in advanced topics and solutions.