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Chapter 3: Problem 41

For Problems \(37-58\), raise each monomial to the indicated power. (Objective 2) $$ \left(-x^{4} y^{5}\right)^{4} $$

Short Answer

Expert verified
\(x^{16} y^{20}\)

Step by step solution

01

Understand the Problem

The problem asks us to raise the entire monomial \((-x^4 y^5)\) to the 4th power.
02

Apply the Power Rule to Each Term

The power rule tells us that \((a^m)^n = a^{m imes n}\). We need to apply this rule to each component inside the parentheses, which are \(-1\), \(x^4\), and \(y^5\).
03

Calculate the Power of the Constant

Raise the constant \(-1\) to the 4th power: \((-1)^4 = 1\) because \(-1\) multiplied by itself an even number of times equals \(1\).
04

Calculate the Power of the x Term

Using the power rule, raise \(x^4\) to the 4th power: \((x^4)^4 = x^{4 imes 4} = x^{16}\).
05

Calculate the Power of the y Term

Raise \(y^5\) to the 4th power: \((y^5)^4 = y^{5 imes 4} = y^{20}\).
06

Combine the Results

Combine the results from each part: \(1 \cdot x^{16} \cdot y^{20} = x^{16} y^{20}\). This is the simplified expression for \((-x^4 y^5)^4\).

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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 a crucial tool in algebra that helps simplify expressions raised to a power. If you see an expression like \((a^m)^n\), you can rewrite it as \(a^{m \times n}\). This rule applies to each part of an expression separately. In simpler terms:
  • Take each component inside the parentheses.
  • Multiply its exponent by the external power.
For example, if you have \(x^4\) and it's all raised to the power of 4, you calculate it as \((x^4)^4 = x^{16}\). The power rule helps break down complex expressions into manageable pieces, making calculations straightforward and systematic.
Exponentiation
Exponentiation is a mathematical operation involving numbers, called bases, raised to a power. An exponent represents how many times the base is multiplied by itself. For example:
  • In \(x^4\), \(x\) is the base, and 4 is the exponent.
  • This means \(x\) is multiplied by itself 4 times: \(x \cdot x \cdot x \cdot x\).
When dealing with negative bases like \(-1\), pay special attention to whether the exponent is even or odd. A negative number raised to an even power becomes positive, while one raised to an odd power stays negative. Using this knowledge, you can handle expressions like \((-1)^4\) effortlessly, knowing it equals 1.
Algebraic Expressions
Algebraic expressions consist of variables, constants, and mathematical operations. Each part of an expression like \(-x^4 y^5\) serves a role:
  • Constants are numbers like \(-1\) that don't change.
  • Variables, such as \(x\) and \(y\), can hold different values.
  • Exponents indicate repeated multiplication.
When working with expressions, it's important to group them correctly. Use parentheses to ensure the right components are being multiplied or raised to powers. Simplifying algebraic expressions often involves applying rules like the power rule to organize and reduce the expression to its simplest form. This turns complex algebraic problems into manageable steps that lead to the final, simplified result like \(x^{16} y^{20}\).

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Most popular questions from this chapter

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 9 x^{4}-37 x^{2}+4=0 $$

Consider the following approach to factoring the problem $$\begin{aligned} &(x-2)^{2}+3(x-2)-10 \\ &(x-2)^{2}+3(x-2)-10 \\ &=y^{2}+3 y-10 \\ &=(y+5)(y-2) \\ &=(x-2+5)(x-2-2) \\ &=(x+3)(x-4) \end{aligned}$$ Use this approach to factor Problems \(110-115\). $$ (x+1)^{2}-8(x+1)+15 $$

How can you determine that \(x^{2}+5 x+12\) is not factorable using integers?

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ -3 x^{2}-19 x+14=0 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ a(a-1)=2 $$
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