Problem 34 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(\frac{v}{w^{-2}}\right)^{-3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{v^3 \cdot w^6}\).

Step 1: Simplify the Base Fraction

The expression is \( \left( \frac{v}{w^{-2}} \right)^{-3} \). We start by simplifying the base fraction. Recall that when you divide by a negative exponent, it is equivalent to multiplying by the reciprocal with a positive exponent. So \( \frac{v}{w^{-2}} \) simplifies to \( v \cdot w^2 \).

Step 2: Apply the Exponent to the Simplified Base

Now, we need to apply the exponent \(-3\) to the expression we obtained from Step 1, \((v \cdot w^2)^{-3}\). Distribute the exponent \(-3\) to both \(v\) and \(w^2\): - \(v^{-3}\) - \((w^2)^{-3} = w^{-6}\).

Step 3: Combine the Exponents

Combine the results from Step 2 to simplify the expression:- The simplified expression is \(v^{-3} \cdot w^{-6}\), which can be rewritten as \(\frac{1}{v^3 \cdot w^6}\) because of the negative exponents.

Key Concepts

Negative Exponents

Negative exponents might seem a bit tricky at first, but they follow a straightforward rule: any number with a negative exponent is equivalent to one divided by that number with a positive exponent.
For example, if you see an expression like \( a^{-n} \), it simplifies to \( \frac{1}{a^n} \).
This transformation is where negative exponents get their power. They allow us to express divisions as multiplications, which can greatly simplify algebraic expressions.
In our exercise, when we come across \( (v \cdot w^2)^{-3} \), each component with a negative exponent simplifies by moving it to the denominator and changing the sign of the exponent. This is why \( v^{-3} \) becomes \( \frac{1}{v^3} \), and \( w^{-6} \) becomes \( \frac{1}{w^6} \). By doing this, we effectively turn complex-looking expressions into something easier to work with.

Reciprocal

To understand how reciprocals play a role in simplifying expressions, think of them as flipping a fraction upside down.
For example, the reciprocal of \( x \) is \( \frac{1}{x} \), and the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
This concept is incredibly useful when dealing with expressions like \( \frac{v}{w^{-2}} \). Instead of getting stuck with the negative exponent in the denominator, we use reciprocals.
By recognizing \( w^{-2} \) as \( \frac{1}{w^2} \), the expression \( \frac{v}{w^{-2}} \) transforms into \( v \cdot w^2 \).
This not only removes the negative exponent but also sets the stage for further simplification.

Exponential Properties

Exponents follow specific properties that make calculations easier. One key property for simplifying expressions is that when raising a power to another power, you multiply the exponents.
This is captured in the rule \( (a^m)^n = a^{m \cdot n} \).
In our example, when the whole expression \( (v \cdot w^2) \) is raised to the power of \(-3\), you apply this property to each part: \( v \) and \( w^2 \).
This leads to \( v^{-3} \) and \( (w^2)^{-3} = w^{-6} \).
Understanding these interactions between exponents not only helps in simplifying expressions but is also a foundational tool for more complex mathematical manipulations.
Additionally, combining these properties can reveal the underlying structure of seemingly difficult expressions, making them easier to manage and understand.