Problem 42 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(x^{20} y^{8}\).

Step 1: Identify the Components of the Expression

The expression given is \[(-x^5 y^2)^4\]where the entire monomial \(-x^5 y^2\) is raised to the 4th power.

Step 2: Apply the Power Rule

When a monomial is raised to a power, apply the power to each factor in the monomial, using the formula \((a^m)^n = a^{m*n}\). Thus,\[(-x^5 y^2)^4 = (-1)^4(x^5)^4(y^2)^4\]which simplifies by applying the powers inside the expression.

Step 3: Solve for Negative Factor

Calculate the power of the negative constant: \[(-1)^4 = 1\]. Since any number raised to an even power becomes positive, the negative sign gets eliminated.

Step 4: Simplify the Variable Parts

Apply the power rule to the variables:\[(x^5)^4 = x^{5*4} = x^{20}\]\[(y^2)^4 = y^{2*4} = y^{8}\].

Step 5: Combine the Results

Combine the results of all parts:\[(-x^5 y^2)^4 = 1 imes x^{20} imes y^{8} = x^{20} y^{8}\]The expression simplifies to \(x^{20} y^{8}\).

Key Concepts

Monomials

Monomials are the simplest types of algebraic expressions and form the building blocks of more complex equations. A monomial comprises a single term which can be a constant number, a variable, or a combination of both multiplied together.
For example, expressions like 7, -3, and -x^2y are all monomials.
Monomials do not feature any addition or subtraction within them, as they are purely multiplicative structures. This simplicity makes them easier to manage, especially in exercises involving exponentiation.

Understanding monomials is crucial for algebra, particularly when performing operations such as multiplication and exponentiation.

• They often involve coefficients, which are the numerical parts like -1 in (-x^5y^2).
• They may contain multiple variables, each potentially raised to a power, as seen in x^5 and y^2.

When working with monomials, it is useful to separate each component (coefficient, base, and exponent) for clarity. This approach helps when applying algebraic rules, such as the power rule, precisely and accurately.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It is written in the form b^n, where b is the base, and n is the exponent. This denotes the base b multiplied by itself n times.
In the case of monomials, exponentiation plays a key role in simplifying expressions.

Consider the expression (-x^5 y^2)^4 from the exercise:

• The base is the monomial, and the exponent is 4.
• Exponentiation requires you to raise every factor within the monomial to the given power.

When exponentiation involves a monomial each part of it is elevated to the power individually.

• For the variable x^5, using the power rule means calculating (x^5)^4, which becomes x^{5*4} = x^{20}.
• For y^2, it's (y^2)^4, resulting in y^{2*4} = y^{8}.

This stepwise method helps to ensure absolute clarity and precision in expanding each part, essential for accurate algebraic manipulation.

Negative Numbers in Algebra

Negative numbers often appear in algebraic expressions, and handling them correctly is essential to avoid errors in calculation. A negative sign in front of a monomial can affect the outcome, especially during exponentiation.

In the example given, (-x^5 y^2)^4 involves a negative sign preceding the entirety of the monomial. The rule here is that when a negative number is raised to an even power, it results in a positive number. This is because multiplying an even amount of negative numbers results in a positive product.

• In the expression, (-1)^4 = 1, which turns negative to positive after exponentiation.
• This means (-x^5 y^2)^4 simplifies to (x^5 y^2)^4, thanks to the power neutralizing the negative sign.

Recognizing how negative numbers transform when exponentiated, especially concerning even and odd powers, helps in simplifying expressions correctly and understanding the deeper mechanics of algebraic computations.