Problem 89 - Algebra for College Students

Question

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(3 x^{n-1}\right)\left(x^{n+1}\right)\left(4 x^{2-n}\right)$$

Step-by-Step Solution

Answer

The product is \(12x^{n+2}\).

Step 1: Break Down the Expression

First, identify each term in the expression. We have \(3x^{n-1}\), \(x^{n+1}\), and \(4x^{2-n}\).

Step 2: Multiply the Coefficients

Start by multiplying the numerical coefficients: 3 and 4. Thus, \(3 \times 4 = 12\).

Step 3: Apply the Laws of Exponents

Apply the law \(x^a \times x^b = x^{a+b}\) to the exponents of \(x\). This combines to \(x^{(n-1) + (n+1) + (2-n)}\).

Step 4: Simplify the Exponent

Inside the exponent, add together \(n-1\), \(n+1\), and \(2-n\). This simplifies to \(n - 1 + n + 1 + 2 - n = n + 2\).

Step 5: Combine Coefficients with Exponents

Combine the result from the previous steps to write the final expression: \(12x^{n+2}\).

Key Concepts

Laws of Exponents

Exponents are a shorthand way of expressing repeated multiplication. When you multiply terms that have the same base, we use a certain set of rules known as the "laws of exponents". These rules tell us how to simplify expressions that involve powers:

• Product of Powers Rule: When you multiply two powers with the same base, you simply add the exponents: \(x^a \times x^b = x^{a+b}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((x^a)^b = x^{a \cdot b}\).
• Power of a Product Rule: When taking a power of a product, apply the exponent to each factor: \((xy)^a = x^a y^a\).

For the exercise, the Product of Powers Rule was directly applied. We see that \(x^{n-1} \times x^{n+1} \times x^{2-n}\) becomes \(x^{(n-1) + (n+1) + (2-n)}\). This shows why understanding these laws is so essential for simplifying complicated expressions.

Multiplying Coefficients

When multiplying terms in an expression, it is important to first address the numerical parts, or coefficients, separately from the variables with exponents. This makes the process simpler and helps avoid confusion:

• Identify all numerical coefficients in each term.
• Multiply these coefficients together. In the problem, we see coefficients 3 and 4 multiplied together to get 12.

This technique allows us to focus solely on simplifying the exponents separately, which can often be the more complex part. By separating the process into manageable steps, you can ensure accuracy and make the problem much less daunting.

Simplifying Exponents

Once you’ve applied the laws of exponents and multiplied the coefficients, the next step is to simplify the expression fully by combining terms. This usually involves working with the exponents:

• Add all the exponents together as outlined by the laws of exponents. In our example, calculate \(n-1 + n+1 + 2-n\).
• Simplify the result, which involves basic arithmetic to combine the like terms.

As seen in this exercise, after adding \(n-1\), \(n+1\), and \(2-n\), you simplify it to get \(n+2\). It's important to ensure all like terms are combined for simplicity. By carefully handling each exponent, you conclude the simplification process with accuracy and clarity, yielding the final result.