Problem 20 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(-\frac{3}{4}x^6y^6\)

Step 1: Identify constants and variables

Identify the constants and variables in each expression. In \( \left(\frac{3}{4} x^{4} y^{5}\right)\), we have a constant \(\frac{3}{4}\) multiplied by variables \(x^4\) and \(y^5\). In \((-x^2 y)\), there is a constant \(-1\) with variables \(x^2\) and \(y\).

Step 2: Multiply the constants

Multiply the constants from each expression: \(\frac{3}{4}\) and \(-1\). This gives \(-\frac{3}{4}\).

Step 3: Apply the product of powers rule to the variables

To multiply variables with the same base, add the exponents. For \(x\), add the exponents: \(x^4 \cdot x^2 = x^{4+2} = x^6\). For \(y\), add the exponents: \(y^5 \cdot y^1 = y^{5+1} = y^6\).

Step 4: Combine results

Combine the constant product with the variable products: \(-\frac{3}{4}x^6y^6\). This is the final product.

Key Concepts

Constants in Algebra

In the world of algebra, constants are fixed values that do not change. Think of them as the numbers without any variables attached. For example, in the expression \(\frac{3}{4}\), \(\frac{3}{4}\) is a constant.
They are crucial because they serve as the numerical part of an algebraic term. Constants can be positive, negative, or even fractions.

• In the expression \(\left(\frac{3}{4} x^{4} y^{5}\right)\), the constant is \(\frac{3}{4}\).
• In the expression \((-x^{2} y)\), the constant is \(-1\).

When multiplying terms, you first deal with constants by multiplying them together. For instance, multiplying \(\frac{3}{4}\) by \(-1\) gives \(-\frac{3}{4}\). Therefore, understanding constants helps simplify and solve expressions efficiently.

Product of Powers Rule

The product of powers rule is a fundamental concept in algebra that helps in simplifying expressions with exponents. If you multiply two expressions that have the same base, you can add their exponents. This rule makes multiplications much simpler when dealing with variables raised to powers.
For example, if you have \(x^a \cdot x^b\), you simply add the exponents \(a\) and \(b\) to get \(x^{a+b}\).
In our given problem, we see this rule in action:

• When you multiply \(x^4 \cdot x^2\), you add the exponents, resulting in \(x^{6}\).
• Similarly, for \(y^5 \cdot y^1\), adding the exponents gives \(y^{6}\).

The product of powers rule simplifies computation and is a vital tool in algebra for combining like terms. It allows for straightforward handling of exponents, making it easier to manage complex expressions.

Variables in Algebra

Variables are symbols used to represent numbers in algebra. They can take many values, and they serve as placeholders or symbols to signify unknowns in equations or expressions. Commonly, variables are denoted by letters such as \(x\), \(y\), and \(z\).
In the expression \(\left(\frac{3}{4} x^{4} y^{5}\right)\), the variables are \(x\) and \(y\). More than just symbols, they reveal the structure of algebraic expressions, indicating how different elements relate to each other.

• The variable \(x\) in this case is raised to the power of 4, showing it is repeated multiplied, \(4\) times.
• For \(y\), since it is raised to the power of 5 in the original expression, it similarly indicates repeated multiplication.

Understanding variables is key to solving equations because they let you express mathematical ideas abstractly, giving a way to solve problems by manipulating these symbols to achieve a desired outcome.