Problem 14 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is \(-x^{4} y^{5}\).

Step 1: Identify bases and exponents

In the expression \((-x^{3} y^{2})(x y^{3})\), identify the bases and their respective exponents. The bases here are \(x\) and \(y\). In the first term \(-x^{3}y^{2}\), \(x\) has an exponent of 3 and \(y\) has an exponent of 2. In the second term \(xy^{3}\), \(x\) has an exponent of 1 and \(y\) has an exponent of 3.

Step 2: Use the Product of Powers Property

Apply the product of powers property, which states that when multiplying two expressions with the same base, you add their exponents. For the base \(x\), add the exponents: \(3 + 1 = 4\). For the base \(y\), add the exponents: \(2 + 3 = 5\).

Step 3: Calculate the product

Multiply the coefficients and write the resulting exponents of each base. The coefficient of the first term is -1, and the second term is 1, so multiply: \((-1) \times 1 = -1\). Combining the results: \((-x^{3} y^{2})(x y^{3}) = -1 \cdot x^{4} y^{5} = -x^{4} y^{5}\).

Key Concepts

Product of Powers Property

The Product of Powers Property is a rule in algebra that helps simplify expressions involving exponents. This property states that when you multiply two expressions that have the same base, you can simply add their exponents together. For example, if you have the expression \( a^m \times a^n \), you can rewrite it as \( a^{m+n} \). This is quite handy because it simplifies the multiplication of powers to a single power with an added exponent.

Let's apply this to our example:

• We have two bases in the expression \((-x^{3} y^{2})(x y^{3})\): \(x\) and \(y\).
• For the base \(x\), you have the exponents 3 and 1. Adding these gives \(3 + 1 = 4\).
• For the base \(y\), the exponents are 2 and 3. Adding these gives \(2 + 3 = 5\).
• So, the product of the powers for \(x\) is \(x^4\) and for \(y\) is \(y^5\).

This property ensures reducing the number of terms, making algebraic expressions easier to work with and quicker to solve. Understanding this property is crucial for solving many mathematical problems involving exponents.

Exponents

Exponents are a key part of algebra, indicating how many times a number, known as the base, is multiplied by itself. If you see an expression like \( x^n \), it reads as "x to the power of n," meaning \(x\) is multiplied by itself \(n\) times.

In our example expression, \((-x^{3} y^{2})(x y^{3})\), we have exponents representing powers:

• For the base \(x\), the exponents are 3 and 1.
• For the base \(y\), the exponents are 2 and 3.

Understanding exponents is essential as they simplify calculations and allow clearer representation of large numbers. They are also the building blocks for applying properties like the Product of Powers.

Knowing how to manipulate these, including multiplying or dividing, is an essential skill in higher mathematics and in practical applications, such as scientific notation and exponential growth.

Coefficients

Coefficients are the numbers in front of variables in algebraic expressions and play a significant role in determining the expression's value. In the given expression \((-x^{3} y^{2})(x y^{3})\), the coefficients are -1 and 1.

When multiplying expressions, it is important to first focus on the coefficients:

• Multiply the coefficients of the two terms: \((-1) \times 1 = -1\). This gives the overall sign and scale of the resulting expression.
• After computing with coefficients, you can proceed with multiplying the variables using their respective exponents.

Coefficients affect the final result, and it's important to handle them correctly to arrive at the right solution. They can also reflect physical quantities in equations used in engineering and sciences, indicating how much of a variable affect things like force, speed, or mass.