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Chapter 0: Problem 44

Simplify each exponential expression. $$\left(-3 x^{4} y^{6}\right)^{3}$$

Short Answer

Expert verified
-27\(x^{12}\)\(y^{18}\)

Step by step solution

01

Applying Exponent Rules

The exponent on the outside of the parenthesis needs to be distributed to each term inside. Thus, \((-3)^3\), \(x^{4*3}\), and \(y^{6*3}\) can be calculated separately.
02

Performing Calculations

Perform these operations: \((-3)^3\) equals -27, \(x^{4*3}\) equals \(x^{12}\), and \(y^{6*3}\) equals \(y^{18}\).
03

Combining the Results

The result can be expressed by combining these calculations. So \((-3)^3 * x^{4*3} * y^{6*3}\) equals -27\(x^{12}\)\(y^{18}\).

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Key Concepts

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

Exponent Rules
Understanding exponent rules is essential when dealing with exponential expressions. These rules simplify the process of manipulating powers in mathematical problems. Let's break down the key rules:
  • Product of Powers Rule: When multiplying like bases, add the exponents: \[ a^m \times a^n = a^{m+n} \]
  • Power of a Power Rule: When raising a power to another power, multiply the exponents:\[ (a^m)^n = a^{m\times n} \]
  • Power of a Product Rule: When raising a product to a power, apply the exponent to each term within the product:\[ (ab)^n = a^n b^n \]
  • Negative Exponents Rule: A negative exponent represents a reciprocal:\[ a^{-n} = \frac{1}{a^n} \]
In our exercise, we applied the "Power of a Power Rule" by distributing the outer exponent to each term inside the parenthesis. This involved multiplying the existing exponents by the outside exponent of 3.
Simplifying Expressions
Simplifying expressions is about breaking down a complex expression into a more manageable form. It's like solving a puzzle by reducing the number of pieces.

Here鈥檚 a simple step-by-step to simplify exponential expressions:
  • Break It Down: If an expression is within parentheses with an exponent outside, apply the exponent to each term inside.
  • Calculate Individually: Deal with each part separately. For numbers, calculate the power. For variables, multiply the exponents.
  • Combine Results: Once each component is simplified, put them back together.
In our example, we first calculated \((-3)^3\), then multiplied the exponents of each variable: \(x^{12}\) and \(y^{18}\). Finally, we combined them into the simplified expression \(-27x^{12}y^{18}\).
Polynomial Expressions
Polynomial expressions consist of variables, exponents, and coefficients. They are often represented as a sum of terms. Each term includes a coefficient multiplied by a variable raised to an exponent.

Key points to understand:
  • Terms: Individual parts of the expression, such as \( -3x^4y^6 \). Each term can have constants and multiple variables.
  • Coefficients: The numerical factor in a term. For instance, \(-3\) in our original term is a coefficient.
  • Degree of a Polynomial: Determined by the highest exponent in any term when the polynomial is expressed in its standard form.
Although the original task was not a full polynomial (since it wasn't a sum of terms), the same principles apply when simplifying powers within polynomial terms.

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Most popular questions from this chapter

Find \(b\) such that \(\frac{7 x+4}{b}+13=x\) will have a solution set given by \(\\{-6\\}\).

This will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\).

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

Perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}-(y+2)^{-1}}{2}$$

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.
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