Problem 57 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(-x y^{4} z^{2}\right)^{7}$$

Step-by-Step Solution

Answer

The simplified expression is \(-x^7 y^{28} z^{14}\).

Step 1: Understand the Power of a Product Rule

The expression given is a monomial inside a set of parentheses raised to a certain power. The exponent applies to each factor inside the parentheses. The rule we use is \[(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n\]Here, the exponent 7 applies to each piece: \(-x\), \(y^4\), and \(z^2\).

Step 2: Apply the Exponent to Each Factor

Apply the exponent 7 to each factor in the monomial. - For \(-x\), it becomes \((-x)^7\).- For \(y^4\), it becomes \((y^4)^7\).- For \(z^2\), it becomes \((z^2)^7\).

Step 3: Simplify Each Factor

Now, simplify each expression individually:- \((-x)^7 = -(x^7)\) since raising a negative number to an odd power keeps it negative.- \((y^4)^7 = y^{4\cdot7} = y^{28}\) by multiplying exponents.- \((z^2)^7 = z^{2\cdot7} = z^{14}\) by multiplying exponents.

Step 4: Combine the Results

Combine the simplified forms of each factor back into a single expression:\[(-x y^4 z^2)^7 = -(x^7) \cdot y^{28} \cdot z^{14}\]So, the simplified expression is \[-x^7 y^{28} z^{14}\].

Key Concepts

Power of a Product Rule

When working with algebraic expressions, especially those that involve exponents, the Power of a Product Rule is a crucial concept to understand. This rule simplifies the process of applying an exponent to a product of terms. Essentially, it states that when you have an expression like \(a \cdot b \cdot c\) raised to a power \(n\), you can apply the exponent individually to each term within the product:

• \( (a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n \)

So, instead of dealing with the whole expression as one unit, you break it down. Each component inside the parentheses gets its own exponent. This rule helps in simplifying expressions effectively and is widely used.
In our original exercise with the expression \((-x y^4 z^2)^7\), the exponent 7 is distributed across all the elements: \(-x\), \(y^4\), and \(z^2\). Applying this rule efficiently breaks down complex algebraic operations.

Monomial

A monomial is a type of algebraic expression that consists of only one term. It can be a single number, a variable, or a product of numbers and variables raised to non-negative integer powers. Monomials are fundamental building blocks in algebra, much like bricks in a wall. They form the basis for constructing more complex expressions like polynomials.

• Examples of monomials include \(5x\), \(7x^2y\), and \(3a^2b^3c^4\).
• A monomial should NOT have addition or subtraction because these operations create multiple terms.

In the given exercise, \((-x y^4 z^2)\) is a monomial expression. It includes three variables multiplied together, each potentially raised to a power. Understanding this concept is key as it helps identify the components to which the Power of a Product Rule can be applied.

Algebraic expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and operations like addition, subtraction, multiplication, and division. They form the language of algebra, enabling us to describe and solve various mathematical problems.

• They can range from simple, like \(x + 5\), to complex, involving exponents or multiple operations like \(3x^2 - 2x + 7\).
• An algebraic expression does not have an "equals" sign - that would make it an equation.

The expression in the original exercise, \((-x y^4 z^2)^7\), is a type of algebraic expression where exponentiation is a part of the operation. Recognizing the components and their interactions, such as how exponents apply to each term, is crucial for simplifying or solving algebra-related problems.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number being multiplied by itself, and the exponent tells us how many times the base is used as a factor in the multiplication. For example, in \(2^3\), 2 is the base, and 3 is the exponent, meaning \(2^3 = 2 \times 2 \times 2 = 8\).

• When an exponent is 1, the base remains unchanged (e.g., \(x^1 = x\)).
• If the exponent is 0, the result is always 1 for any non-zero base (e.g., \(x^0 = 1\)).

In expressing and simplifying algebraic expressions, recognizing how exponentiation works with rules, such as the Power of a Product Rule, is essential. In our exercise, aspects like distributing the power across a product and multiplying powers (\((b^m)^n = b^{m \cdot n}\)) play an integral role in the simplification process. Understanding this concept aids in navigating through increasingly complex algebraic problems with ease.