Problem 123 - Elementary and Intermediate Algebra

Question

Perform each operation. $$ 3 x^{2} y^{3}\left(-5 x^{3} y^{-4}\right) $$

Step-by-Step Solution

Answer

\(-\frac{15x^5}{y}\)

Step 1: Understand the operation

The expression given involves the multiplication of two monomials. We need to multiply the coefficients and apply the properties of exponents to the variable parts.

Step 2: Multiply the coefficients

Identify and multiply the numerical coefficients of the monomials. Here, the coefficients are 3 and -5.Thus: \(3 imes (-5) = -15.\)

Step 3: Use the product of powers property for variables

When multiplying monomials, add the exponents for like bases. For \(x\): Add the exponents: \(2 + 3 = 5\).For \(y\): Add the exponents: \(3 + (-4) = -1\).

Step 4: Write the resulting expression

Combine the results from steps 2 and 3:\(-15x^5y^{-1}\).

Step 5: Simplify if needed

The expression can be simplified to use only positive exponents:\(-15x^5 \frac{1}{y}\).

Step 6: Finalize the expression

The final simplified expression is:\(-\frac{15x^5}{y}\).

Key Concepts

Exponents

Exponents are a fundamental concept in algebra, playing a crucial role in polynomial operations like multiplication. When you have an expression such as \(x^2\), the number 2 is the exponent, indicating that \(x\) is multiplied by itself 2 times: \(x \times x\). The operation of exponents includes properties that help us simplify expressions efficiently.

• Product of Powers: When multiplying two expressions with the same base, such as \(x^m \times x^n\), you simply add the exponents: \(x^{m+n}\). This is why in the original exercise, for the \(x\) terms, we add 2 and 3 to get \(x^5\).
• Negative Exponents: A negative exponent implies a reciprocal. For example, \(y^{-1}\) is the same as \(\frac{1}{y}\). In the expression \(-15x^5y^{-1}\), \(y^{-1}\) signifies division by \(y\).

Understanding these properties helps in manipulating and simplifying expressions with ease.

Monomials

A monomial is an algebraic expression with only one term. It could be a constant, a variable, or a product of constants and variables. The expression \(3x^2y^3\) is a monomial because it contains only one term made by multiplying numerical (3) and variable parts (\(x^2y^3\)).

• Coefficients: This is the numerical factor in a term. For example, in \(3x^2y^3\), the coefficient is 3. It's the constant multiplying the variables.
• Variables and Exponents: Variables are the letters in a term, and exponents show how many times the variable is used as a factor. In \(x^2\), \(x\) is the variable, and 2 is the exponent.

When multiplying monomials, multiply the coefficients and apply the rules of exponents to the variable parts.

Simplifying Expressions

Simplifying expressions is the process of making an expression easier to understand or work with. This often involves reducing terms to their simplest form, especially in polynomial expressions involving exponents.

• Combining Like Terms: This means summing up terms with the same variable and the same exponents. It is more applicable in addition or subtraction of polynomials, but the principle of dealing with the same bases is crucial in multiplication too.
• Rewriting Negative Exponents: This involves converting negative exponents into fractions. For instance, rewriting \(y^{-1}\) as \(\frac{1}{y}\) simplifies the expression further to be more conventional.

In our exercise, after multiplying and applying the product of powers rule, we simplify the expression by converting negative exponents and writing it as \(-\frac{15x^5}{y}\). This final form is more precise and universally understood.