Problem 58 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(-x^{2} y^{4} z^{5}\right)^{5}$$

Step-by-Step Solution

Answer

The expression simplifies to \(-x^{10} y^{20} z^{25}\).

Step 1: Apply the Power of a Monomial Rule

To raise a monomial to a power, apply the rule: \[ (a^m b^n c^p)^q = a^{mq} b^{nq} c^{pq} \] In this problem, we have \((-x^2 y^4 z^5)^5\). Apply this rule to each component.

Step 2: Apply the Power Rule to Each Component

First, apply the power rule to \(-1\), since the sign affects the final answer: \[ (-1)^5 = -1 \] Now, apply the power rule to each variable: \[ x^{2*5} = x^{10} \] \[ y^{4*5} = y^{20} \] \[ z^{5*5} = z^{25} \] Combine these to form the result.

Step 3: Combine the Results

The expression is now: \[ -1 imes x^{10} imes y^{20} imes z^{25} \] This simplifies to: \[ -x^{10} y^{20} z^{25} \] This is the final simplified expression.

Key Concepts

Exponentiation

Exponentiation is the mathematical operation where a number, known as the base, is raised to a power. This power, or exponent, indicates how many times the base is multiplied by itself. For instance, in the expression \(5^3\), the base 5 is multiplied by itself three times, resulting in \(5 \times 5 \times 5 = 125\). Similarly, if we have an expression like \((-x^{2} y^{4} z^{5})^{5}\), each part of the expression is raised to the power of 5.
By understanding exponentiation, you can simplify complex problems. Here are some basic rules:

• Product Rule: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^n = a^n \times b^n\)

These rules help break down more difficult expressions into manageable parts, thus making calculations simpler.

Power of a monomial

The power of a monomial involves raising each component of a monomial expression to a given exponent. A monomial is an algebraic expression made up of numbers and variables multiplied together, such as \(-x^2 y^4 z^5\). When we raise a monomial to a power, like \((-x^2 y^4 z^5)^5\), we must apply the exponent to every part of the expression.Here is how it is typically done:

• Apply the power to the numerical coefficient. For instance, \((-1)^5 = -1\).
• Apply the power to each variable by multiplying the inside exponent by the outer exponent: \(x^{2 \times 5} = x^{10}\), \(y^{4 \times 5} = y^{20}\), and \(z^{5 \times 5} = z^{25}\).

Once you have applied the power to each component, you recombine them to form the final expression, like \(-x^{10}y^{20}z^{25}\) in the original problem.

Simplifying expressions

Simplifying expressions is the process of making expressions easier to work with by condensing them into their simplest forms. This is analogous to reducing a fraction to its lowest terms. In algebra, whether dealing with monomials or polynomials, simplifying often involves combining like terms and applying rules of exponentiation effectively.Consider the expression \((-x^{2} y^{4} z^{5})^{5}\). Simplifying involves raising each term to the power of 5, and then combining them into a single expression. As shown in the solution, you first calculate each powered component:

• \((-1)^5 = -1\)
• \(x^{2 \times 5} = x^{10}\)
• \(y^{4 \times 5} = y^{20}\)
• \(z^{5 \times 5} = z^{25}\)

Once computed, these terms are multiplied together to create the fully simplified monomial \(-x^{10} y^{20} z^{25}\). This practice not only makes expressions manageable but also lays a foundation for solving more complex algebraic problems.

Algebraic expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operators such as addition, subtraction, multiplication, and division. Unlike simple arithmetic expressions, algebraic expressions can contain variables that represent numbers, and these are often used in equations describing real-world phenomena.For instance, consider the algebraic expression \(-x^{2} y^{4} z^{5}\). This expression includes variables \(x\), \(y\), and \(z\) with associated powers, representing the relationship between these quantities in product form. When raising an algebraic expression to a power, such as \((-x^{2} y^{4} z^{5})^{5}\), each part of the expression is raised to that power as demonstrated in the problem solution.Understanding algebraic expressions and operations on them is crucial because:

• They represent real-world problems mathematically.
• They are foundational for learning more complex algebraic concepts.
• They provide the groundwork for graphing, solving equations, and modeling situations in calculus and other advanced math subjects.

Thus, mastering algebraic expressions enables students not only to excel in mathematics but also to apply these skills in scientific and engineering domains.