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Chapter 5: Problem 93

Simplify each expression. See Examples 1 through 11. $$ \left(-6 x y z^{3}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \(36x^2y^2z^6\).

Step by step solution

01

Understand the Expression

We need to simplify the expression \((-6xyz^3)^2\). This means the entire term \(-6xyz^3\) is being squared.
02

Square Each Component

Square each part of the term individually: the numerical coefficient, and each variable with its exponent.- Square the coefficient: \((-6)^2 = 36\).- Square each variable: - \(x^2\) (because \(x\) raised to the power of 1 is individually squared) - \(y^2\) - \((z^3)^2 = z^{6}\) because you multiply the exponents when raising a power to another power.
03

Construct the Simplified Expression

Combine all squared components to form the full simplified expression: \(36x^2y^2z^6\).

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation that involves raising a base to the power of an exponent. In simple terms, it's like multiplying a number by itself a certain number of times. When you see an expression like
  • \((-6xyz^3)^2\)
, it indicates that everything inside the parenthesis is raised to the power of 2.
This means you have to multiply the whole expression
  • \((-6xyz^3)\)
by itself. When dealing with exponentiation, it's crucial to remember rules like:
  • squaring a negative number results in a positive one, as \((-6) imes (-6) = 36\).
  • For powers of powers, multiply the exponents together.
For example, in \((z^3)^2\), you multiply 3 and 2 to get \(z^6\). Exponentiation simplifies complex expressions by reducing repeated multiplication into easier terms.
Variables
Variables are symbols used in mathematics to represent unknown or changeable values. In the expression
  • \((-6xyz^3)^2\)
, the characters
  • \(x\), \(y\), \(z\)
are variables. They act as placeholders for numbers and can change depending on the context or given information.
Variables are fundamental in algebra as they help form equations and expressions that model real-world phenomena. When simplifying expressions, treat each variable with its given exponent separately, like transforming
  • \(x^1\) into \(x^2\) by squaring it.
  • Similarly, \(y^1\) becomes \(y^2\) and \((z^3)^2\) turns into \(z^6\).
In algebraic operations, variables are key to forming equations you can manipulate and solve.
Mathematical Expressions
Mathematical expressions combine numbers, variables, and operations to convey a calculation or quantity. The task of simplification involves making these expressions easier to understand and solve. For the expression
  • \((-6xyz^3)^2\)
, it involves:
  • Breaking down each part inside the parenthesis.
  • Applying operations like squaring to each component 鈥� coefficient and variables.
This expression is constructed from multiple elements: a negative coefficient, variables, and exponents. Simplifying such expressions follows certain rules, like removing parentheses by distributing the square to all terms, resulting in
  • \(36x^2y^2z^6\).
When dealing with complex expressions, it's about systematically applying algebraic rules to reduce the expression to its simplest form, making it easier to work with or solve in subsequent mathematical procedures.

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