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Chapter 4: Problem 106

Simplify. $$\left(x y z^{3}\right)\left(x y^{2} z^{2}\right)^{3}$$

Short Answer

Expert verified
The simplified form of the expression is \(x^{4} y^{7} z^{9}\)

Step by step solution

01

Apply the Cube Power to Each Term

The cube power applies to each of the terms inside the brackets. We obtain: \( x y z^{3} \times x^{3} y^{6} z^{6} \)
02

Combine like terms

According to the laws of exponents, to multiply terms with the same base, add the exponents. First, combine the \(x\) terms, then the \(y\) terms and lastly the \(z\) terms. This gives us: \(x^{1+3} y^{1+6} z^{3+6}\)
03

Simplify

The expression simplifies to: \(x^{4} y^{7} z^{9}\)

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Key Concepts

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

Laws of Exponents
The laws of exponents are fundamental rules that help simplify expressions involving powers. When you multiply terms with the same base, simply add their exponents together. For example, when combining \(x^a\) and \(x^b\), you get \(x^{a+b}\).
This principle is crucial when dealing with expressions like \((xy^2z^2)^3\). Here, we apply the cube power to each element inside the brackets, yielding \(x^3y^6z^6\) because each exponent is multiplied by 3.
Employing these laws makes it much easier to manage complex expressions and ensures that your calculations remain consistent.
Polynomials
Polynomials are algebraic expressions consisting of variables, coefficients, and exponents that are combined using addition, subtraction, and multiplication. They can range from a single term, known as a monomial, to multiple terms, such as binomials and trinomials.
In our exercise, the expression \(xy^2z^2\) is a monomial, and when raised to a power, each part of the term is independently affected by the exponent, leading to several calculations.
This manipulation allows us to break down even the most difficult expressions into manageable pieces, ultimately making calculations more straightforward.
  • Monomials: Single-term polynomials like \(3x^2\)
  • Binomials: Two-term polynomials such as \(x + y\)
  • Trinomials: Three-term polynomials like \(x^2 - 4x + 4\)
Mathematical Operations
Mathematical operations are the basic building blocks of algebra, and they include addition, subtraction, multiplication, and division. Each operation has specific rules that define how numbers and variables interact with each other.
In constructing and simplifying expressions:
  • Addition/Subtraction: These operations involve combining like terms, such as adding or subtracting coefficients of variables with the same exponents.
  • Multiplication: Involves using the distributive property and the laws of exponents to combine terms. For instance, multiplying \((xy)\) by \((y^2z)\) involves a systematic approach where each term interacts with every other corresponding term.
  • Division: Essentially the inverse of multiplication, division of expressions uses cancellation and simplification tactics.

Using these operations efficiently lets you simplify expressions and solve algebraic equations effectively. They provide a structured way to approach problems and ensure rigorous accuracy in solutions.

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Most popular questions from this chapter

Find the difference when \(\left(-3 z^{3}-4 z+7\right)\) is subtracted from the sum of \(\left(2 z^{2}+3 z-7\right)\) and \(\left(-4 z^{3}-2 z-3\right)\)

Classify each polynomial as a monomial, a binomial, a trinomial, or none of these. $$\frac{1}{2} x^{3}+3$$

Classify each polynomial as a monomial, a binomial, a trinomial, or none of these. $$12 x^{3}-12 x^{2}+36 x-3$$

Graph each polynomial function. State the domain and range. $$f(x)=x^{3}+2$$

Perform the operations. $$5 x^{5} y^{10}-\left(2 x y^{2}\right)^{5}+(3 x)^{5} y^{10}$$
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