Problem 77 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \left(\frac{3 t^{2}}{2 t^{-1}}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{27}{8} t^9 \).

Step 1: Apply the Power of a Quotient Rule

First, apply the power of a quotient rule, which states \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). Therefore, we have: \[ \left(\frac{3t^2}{2t^{-1}}\right)^3 = \frac{(3t^2)^3}{(2t^{-1})^3} \]

Step 2: Apply the Power of a Product Rule

Now, apply the power of a product rule, which states \( (ab)^n = a^n b^n \). For the numerator \((3t^2)^3\), distribute the exponent across both the 3 and \(t^2\): \[ (3t^2)^3 = 3^3 (t^2)^3 = 27t^6 \] For the denominator \((2t^{-1})^3\): \[ (2t^{-1})^3 = 2^3 (t^{-1})^3 = 8t^{-3} \]

Step 3: Simplify Using Quotient Rule for Exponents

Use the quotient rule for exponents, which states \( \frac{a^m}{a^n} = a^{m-n} \), to simplify \( \frac{t^6}{t^{-3}} \): \[ t^{6 - (-3)} = t^{6 + 3} = t^9 \] Now the expression becomes: \[ \frac{27t^6}{8t^{-3}} = \frac{27}{8} t^9 \]

Step 4: Final Simplified Expression

Combine everything to get the final answer with only positive exponents: \[ \frac{27}{8} t^9 \]

Key Concepts

Power of a Quotient Rule

When you come across a fraction inside an exponent, you can simplify it by using the **Power of a Quotient Rule**. This rule is like a handy shortcut that tells you that raising a fraction to a power is the same as raising both the top and bottom of the fraction to that power separately. Think of it as spreading the power to each part of the fraction.

• Mathematical Formula: For any numbers or variables, it is expressed as \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \).
• Application: In solving \( \left( \frac{3t^2}{2t^{-1}} \right)^3 \), this turns the expression into \( \frac{(3t^2)^3}{(2t^{-1})^3} \).

Using this rule early on saves you from much confusion, allowing each component of the fraction to handle their own exponents separately.

Power of a Product Rule

The **Power of a Product Rule** is your go-to option when you have multiple terms multiplied together and all of them are raised to the same power. It's like the power is shared equally among the numbers or variables being multiplied.

• Mathematical Formula: For any numbers or variables, \( (ab)^n = a^n b^n \).
• Application in Example: Using \((3t^2)^3\), distribute the 3 as \(3^3 (t^2)^3 = 27t^6\).
• Similarly, in \((2t^{-1})^3\), apply the product rule as \(2^3 (t^{-1})^3 = 8t^{-3}\).

Each term in the original product gets its own power, simplifying the product into more manageable pieces. This helps break down complex expressions into simpler components quickly.

Quotient Rule for Exponents

Now, let's talk about the **Quotient Rule for Exponents**. It comes in handy when you have the same base raised to exponents and divided by one another. It's like a more efficient shortcut to reducing the power of the bottom from the top.

• Mathematical Formula: If you have \( \frac{a^m}{a^n} = a^{m-n} \), this is your rule in play.
• Application in Example: When simplifying \( \frac{t^6}{t^{-3}} \), you subtract the exponents: \(t^{6 - (-3)} = t^{6 + 3} = t^9\).

This rule streamlines expressions, particularly when you deal with variables having powers on both sides of a division. By subtracting exponents, you quickly find the simplified form, as seen in our example yielding \( \frac{27}{8} t^9 \). This simplification step is key to maintaining clarity and accuracy in exponentiation tasks.