Problem 54 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$-\left(x y^{2} z^{3}\right)^{8}$$

Step-by-Step Solution

Answer

The result is \(-x^8 y^{16} z^{24}\).

Step 1: Identify the Components of the Monomial

The given monomial is \(-\left( x y^{2} z^{3} \right)^{8}\). It comprises a negative sign and three variables: \(x\), \(y^2\), and \(z^3\). Our goal is to raise each part to the 8th power.

Step 2: Apply the Power to the Monomial

We'll apply the power to each component inside the parentheses. The expression can be rewritten as:\(- (x^{1})^8 (y^{2})^8 (z^{3})^8\).

Step 3: Calculate Each Power

This step involves raising each base to the given power. We perform the calculations as follows:- \(x^{1 \times 8} = x^{8}\)- \(y^{2 \times 8} = y^{16}\)- \(z^{3 \times 8} = z^{24}\)

Step 4: Apply Negative Sign

Since the original expression has a negative sign outside the parentheses, it applies to the final product after the powers have been calculated, resulting in the expression being negative:\(- x^8 y^{16} z^{24}\).

Key Concepts

Monomial

A monomial is a single-term expression in algebra that consists of a constant, variables, or both multiplied together. It is important to recognize that a monomial does not include any addition or subtraction operators. For instance, terms like \(4x^3\), \(-y^{2}\), and \(xyz\) are all examples of monomials. They can only have whole number exponents associated with their variables.
Understanding monomials is crucial when you're dealing with expressions that will undergo operations such as exponentiation. In our exercise, the monomial \[-\left( x y^{2} z^{3} \right)\]is raised to a power. Hence, recognizing the structure of a monomial helps in efficiently applying mathematical rules, like those involved with powers, to simplify and solve algebraic expressions.

Power Rule

The power rule is an essential rule in algebra that simplifies the process of raising terms to a power. It states that when you raise a power to another power, you multiply the exponents.
This rule applies whether you are working with one variable or several variables within a term. Mathematically, the power rule can be expressed as: - For a single term, \((x^m)^n = x^{m \times n}\)
This means you take the exponent \(m\) and multiply it by \(n\) to get the new exponent.
In the exercise solution, the power rule is crucial when calculating each component inside the parentheses. Each base variable (such as \(x\), \(y^2\), \(z^3\)) is raised to the 8th power by multiplying the existing powers by 8:

• \(x^{1 \times 8} = x^{8}\)
• \(y^{2 \times 8} = y^{16}\)
• \(z^{3 \times 8} = z^{24}\)

Utilizing the power rule reduces complex expressions into simpler, manageable terms, making further algebraic operations more straightforward.

Negative Sign Handling

Handling a negative sign in mathematical expressions is straightforward yet important, especially as it can change the outcome of calculations. When a negative sign precedes an expression or a monomial, it indicates that the entire result should be negated.
In the exercise given \[-\left( x y^{2} z^{3} \right)^{8}\], the negative sign outside the parentheses means that after calculating the powers of each term, we multiply the results by \(-1\). During exponentiation, the negative sign is treated independently from the powers applied to the variables within the monomial, and it applies to the final product of the raised expression.
As outlined in the solution, after computing the powers:

• \(x^{8}\)
• \(y^{16}\)
• \(z^{24}\)

The expression is finalized by applying the negative sign: \(- x^8 y^{16} z^{24}\).This demonstrates not only the necessity of handling the negative sign correctly but also its capacity to affect the overall sign of the expression's product.