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Chapter 6: Problem 78

Simplify. Do not use negative exponents in the answer. $$ \left(-5 x^{2} y^{-6}\right)^{-3} $$

Short Answer

Expert verified
-\frac{y^{18}}{125x^6}

Step by step solution

01

Apply the Power Rule

Use the power rule for exponents \((a^m)^n = a^{mn}\) to distribute the exponent \(-3\) to each part of the expression. \[ \begin{align*} \left( -5 x^2 y^{-6} \right)^{-3} &= (-5)^{-3} (x^2)^{-3} (y^{-6})^{-3} \end{align*} \]
02

Simplify Each Component

Evaluate each part separately. For the constants and variables, apply the power rule \(a^m * a^n = a^{m+n}\). \[ (-5)^{-3} = -\frac{1}{5^3} = -\frac{1}{125}, \ (x^2)^{-3} = x^{2(-3)} = x^{-6}, \ (y^{-6})^{-3} = y^{-6(-3)} = y^{18} \]
03

Combine and Rewrite without Negative Exponents

Now, combine the simplified components and rewrite any parts with negative exponents. In this case, move the terms with negative exponents to the denominator. \[ -\frac{1}{125} \cdot x^{-6} \cdot y^{18} = -\frac{y^{18}}{125x^6} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is essential for simplifying expressions with exponents. It states that when you have an exponent raised to another exponent, you multiply the exponents. This can be written as \((a^m)^n = a^{m \times n}\).
In the given exercise, \((-5 x^2 y^{-6})^{-3}\), we use the power rule to apply the exponent \-3\ across each component.
This simplifies our expression, eventually breaking it down into separate parts that are easier to handle. Each component's exponent is multiplied by \-3\, transforming our complex expression into a series of simpler calculations.
Keep in mind this rule applies to every factor inside the parentheses.
Negative Exponents
Negative exponents can initially seem confusing, but they follow a straightforward rule: A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.
For instance, \(a^{-n} = \frac{1}{a^n}\).
In the solution step \((-5)^{-3} = - \frac{1}{5^3}\), we converted a negative exponent into its reciprocal form.
Similarly, for \(x^{-6}\), another term in the expression, we can rewrite this as \(\frac{1}{x^6}\).
Understanding and rewriting negative exponents is essential for simplifying the expression and ensure our final answer avoids any negative exponents.
Exponentiation
Exponentiation involves raising a base to a certain power. For example, \(a^n\) means multiplying \a\ by itself \ times.
In our exercise, we raised each part of \(-5 x^2 y^{-6}\) to \-3\, applying the power rule. Let's break it down:
  • \((-5)^{-3}\) simplifies to amn \(-\frac{1}{125}\). This follows from multiplying \(-5\) three times since \-3\ is our exponent.
  • \((x^2)^{-3} = x^{-6}\). Here, each \x\ in \x^2\ is raised to \-3\, resulting in \-6\ as our new exponent.
  • \((y^{-6})^{-3}\) yields \(y^{18}\) since multiplying two negative exponents results in a positive one.

Understanding these exponentiation steps allows us to simplify complex expressions accurately. It鈥檚 a building block for algebra that aids in everything from simple equations to advanced calculus.

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