Problem 9 - Algebra A Combined Function

Question

Multiply. \(\left(-\frac{1}{3} y^{2}\right)\left(\frac{2}{5} y\right)\)

Step-by-Step Solution

Answer

The product is \(-\frac{2}{15} y^3\).

Step 1: Multiply the Constants

First, we multiply the coefficients of the two expressions. The coefficients are \(-\frac{1}{3}\) and \(\frac{2}{5}\). Multiplying them gives: \[-\frac{1}{3} \times \frac{2}{5} = -\frac{2}{15}\].

Step 2: Multiply the Variables

Next, multiply the variables in the expressions. We have \(y^2\) and \(y\), which when multiplied give \(y^{2+1} = y^3\).

Step 3: Combine Results

Combine the results from Step 1 and Step 2. The product of the expressions is:\[-\frac{2}{15} y^3\].

Key Concepts

Multiplying Coefficients

When you multiply coefficients, you're simply multiplying the numerical parts of the algebraic terms. In this exercise, the coefficients given are \(-\frac{1}{3}\) and \(\frac{2}{5}\). Think of coefficients as the multipliers of the variable parts of your expression.
Here's how you do it step-by-step:

• You multiply the numerators together: \(-1 \times 2 = -2\).
• Next, multiply the denominators: \(3 \times 5 = 15\).

So, the product of the coefficients is \(-\frac{2}{15}\).
Remember, keep track of the negative signs; in this case, a negative times a positive gives a negative product.

Exponent Rules

Exponent rules help you simplify expressions that involve powers. When multiplying terms that have the same base, you can add the exponents together. In our problem, we have \(y^2\) and \(y\). These are like terms in terms of their base.
Here’s what you do:

• Write the expression using its exponential form: the single \( y \) is actually \( y^1 \).
• Add the exponents for the common base: \( 2 + 1 = 3 \).

Therefore, \( y^2 \times y \) results in \ y^3 \. This is because you’re technically multiplying \ y \ by itself three times.

Combine Like Terms

Combining like terms is the process of consolidating terms in an expression that have the same variable raised to the same power.
In our example, after multiplying the coefficients and applying the exponent rules, we're left with \(-\frac{2}{15} y^3\). Since there are no other \(y^3\) terms to combine with, this is our final expression.
Here are some key points to remember:

• Only terms with exactly the same variable and exponent can be combined.
• Ensure all like terms are simplified together to get a cleaner final result.

Even though we didn’t have to combine any terms in this exercise, it's a good principle to apply when working with longer and more complex expressions.