Problem 16 - Algebra for College Students

Question

Find each product. $$\left(\frac{1}{2} x^{2} y^{6}\right)\left(\frac{2}{3} x y\right)$$

Step-by-Step Solution

Answer

\( \frac{1}{3} x^3 y^7 \)

Step 1: Multiply the Constants

Start by multiplying the constant coefficients from both expressions. In this case, we have \( \frac{1}{2} \) from the first expression and \( \frac{2}{3} \) from the second expression. To multiply these, compute \( \frac{1}{2} \times \frac{2}{3} = \frac{1 \cdot 2}{2 \cdot 3} = \frac{2}{6} = \frac{1}{3} \).

Step 2: Multiply the Like Variables

Next, multiply the like variables from both expressions. For \( x \), we have \( x^2 \) in the first expression and \( x \) in the second expression. Multiply these to get \( x^{2+1} = x^3 \).

Step 3: Continue with Other Variables

For variable \( y \), we have \( y^6 \) in the first expression and \( y \) in the second expression. Multiply these to get \( y^{6+1} = y^7 \).

Step 4: Write the Final Product

Combine the results from Steps 1, 2, and 3 to write the final expression: \( \frac{1}{3} x^3 y^7 \).

Key Concepts

Constant Multiplication

In algebraic multiplication, dealing with constant multiplication is often the first step in simplifying expressions. Constants are the standalone numbers that are not attached to variables. When multiplying constants, you simply multiply the numbers as you would in elementary arithmetic. This involves the following steps:

• Identify the constants in each expression. This involves looking for numbers that are not multiplied by variables. For example, in the expression \( \left(\frac{1}{2} x^{2} y^{6}\right)\), the constant is \( \frac{1}{2} \).
• Multiply these constants together. For our problem, you take \( \frac{1}{2} \) and \( \frac{2}{3} \), then compute their product: \( \frac{1}{2} \times \frac{2}{3} = \frac{2}{6} \), which simplifies to \( \frac{1}{3} \).

Understanding this process helps set the foundation for what follows in the multiplication of the entire expression. Simplifying fractions, as shown above, is also a crucial part of constant multiplication.

Exponent Rules

Exponent rules are essential when dealing with powers during multiplication. Two important rules to remember are the product of powers rule and the power of a power rule. Here's how you apply them:

• Product of Powers Rule: When multiplying two powers with the same base, you add their exponents. If you have \( x^a \times x^b \), you simplify it to \( x^{a+b} \).
• Applying to Variables: For the variable \( x \) in our problem, with \( x^2 \) in the first expression and \( x \) in the second, you apply the rule: \( x^{2+1} = x^3 \).

For the variable \( y \), which appears as \( y^6 \) and \( y \), you apply the product of powers rule: \( y^{6+1} = y^7 \). Being comfortable with these rules allows for quick simplification of expressions involving exponents.

Variable Multiplication

When multiplying expressions with variables, consider both the coefficients and how the same variables' powers interact. Here’s a simple guide:

• Multiply constants as described in constant multiplication.
• Use exponent rules to multiply like variables. Add the exponents if the base is the same, as discussed.
• Identify unique variables separately and apply the rules accordingly.

For instance, in our problem, after determining \( \frac{1}{3} \) from constant multiplication, we know \( x^3 \) is from multiplying \( x^2 \) and \( x^1 \), and \( y^7 \) from \( y^6 \) and \( y^1 \). Combine these results to form the complete product: \( \frac{1}{3} x^3 y^7 \). Understanding variable multiplication helps in correctly aligning all parts of the expression.