Problem 27 - Algebra for College Students

Question

Find each product. $$\left(-3 y^{2}\right)\left(-2 y^{2}\right)\left(-4 y^{5}\right)$$

Step-by-Step Solution

Answer

The product is \(-24y^9\).

Step 1: Multiply the Constants

First, look at the numerical coefficients in each parenthesis: (-3), (-2), and (-4). Multiply these constants together: \(-3 \cdot -2 \cdot -4\). Calculate: \[(-3) \times (-2) = 6\]\[6 \times (-4) = -24\].So, the product of the constants is \(-24\).

Step 2: Multiply the Variables

Combine the like terms that include the variable \(y\). Multiply \(y^2\), \(y^2\), and \(y^5\) by adding their exponents:y-th powers: \(y^{2+2+5} = y^9\).Hence, the multiplication of the variables gives \(y^9\).

Step 3: Combine Everything Together

Combine the product of the constants and the product of the variables. We have:\[-24 \times y^9 = -24y^9\]Therefore, the final product of the expression \(\left(-3 y^{2}\right)\left(-2 y^{2}\right)\left(-4 y^{5}\right)\) is \(-24 y^9\).

Key Concepts

Exponentiation

Exponentiation is the process of multiplying a number by itself a certain number of times.
In the expression \(-3y^2\), the exponent is 2, which means \(y\) is multiplied by itself: \(y \times y\). In problems involving multiplication of variables with exponents, such as \(y^2\) and \(y^5\), the key is to add the exponents when the bases are the same.
This is known as the product of powers property, which can be written as \(a^m \times a^n = a^{m+n}\).
It simplifies the multiplication of two numbers that have the same base. A quick recap on how this works:

• When multiplying \(y^2 \times y^2\), we add the exponents: \(2 + 2 = 4\).
• Next, multiplying the result \(y^4\) by \(y^5\), we add the exponents again: \(4 + 5 = 9\).

In our example, when you combine all the \(y\)'s together, you get \(y^9\).
Understanding this process helps make seemingly complex algebraic expressions much more manageable.

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in the other polynomial.
In simpler terms, it's all about distributing and combining like terms properly.
While our exercise doesn't have multiple terms (since each parenthesis is a monomial with only one term), understanding how polynomial multiplication works will be helpful in more complex problems. An easy checklist when handling polynomial multiplication:

• Multiply the coefficients. For instance, in the exercise: \((-3), (-2), (-4)\).
• Multiply the variables using exponent rules. As seen with \(y^2 \times y^2 \times y^5\).

Remember, careful multiplication and addition of exponents will allow you to correctly combine your like terms, resulting in the expression's product. This technique ensures precision and accuracy in outcomes.

Numerical Coefficients

Numerical coefficients are the numbers that appear in front of variables in algebraic expressions.
In this exercise, the coefficients are -3, -2, and -4. When dealing with such expressions, the coefficients are multiplied separately from the variables.
This is a straightforward arithmetic operation.Here's how to handle them:

• Identify and list out the coefficients: \(-3, -2, -4\).
• Multiply them in sequence: first, \((-3) \times (-2) = 6\).
• Continue multiplying the product by the next coefficient: \(6 \times (-4) = -24\).

The resulting coefficient, \(-24\), also impacts the overall sign of the product.
The number of negative signs determines if the final product is positive or negative.
Here, an odd number of negative signs results in a negative product: \(-24\). Understanding this process aids in organizing and solving algebraic expressions efficiently.