Problem 21 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is \(30x^6\).

Step 1: Multiply the Numerical Coefficients

First, identify and multiply the numerical coefficients. The coefficients in this expression are 3, -2, and -5. Multiply them together: \[3 \times (-2) \times (-5)\]Simplify this expression to get:\[3 \times (-2) = -6\]\[-6 \times (-5) = 30\]So, the product of the numerical coefficients is 30.

Step 2: Add the Exponents of x

Next, add the exponents for each 'x' term. The expression has \(x^1\), \(x^2\), and \(x^3\). We add the exponents as follows:\[1 + 2 + 3 = 6\]Thus, the resulting power of \(x\) will be \(x^6\).

Step 3: Combine the Results

Combine the result of Step 1 and Step 2. The combined result gives:\[30 \times x^6\]Therefore, the product of the entire expression is:\[30x^6\]

Key Concepts

Numerical Coefficients

When multiplying monomials, the first step is to focus on the numerical coefficients. In algebra, the numerical coefficient is the number that is multiplied by the variable or variables. Identifying these coefficients is simple and vital. For our exercise, the coefficients are 3, -2, and -5.

• First, take 3 and multiply it by -2. This results in -6.
• Then, take that -6 and multiply it by -5 to get 30.

Notice that multiplying a negative by a negative yields a positive result. Thus, understanding and correctly multiplying numerical coefficients is a key part of solving multiplication problems involving monomials.

Exponents

Exponents indicate how many times you multiply a number by itself. In our problem, each 'x' term has an exponent, even if it is not immediately visible.

• The term \(3x\) has an exponent of 1 on the \(x\), so it is \(x^1\).
• The term \(-2x^2\) already shows an exponent of 2.
• Lastly, the term \(-5x^3\) highlights the exponent of 3.

To find the power of \(x\) in the solution, add these exponents together: \(1 + 2 + 3 = 6\). Hence, the combined power of \(x\) in the final expression is \(x^6\). Adding exponents correctly allows for simplifying the expression effectively.

Product of Powers

The "product of powers" rule states that when you multiply powers with the same base, you can simply add the exponents. This is super helpful when dealing with expressions like in our exercise. By reflecting on our example:

• We applied the rule by adding the exponents \(1 + 2 + 3 = 6\).
• This is why our solution has \(x^6\) as the combined power of \(x\).

Remember, this rule only applies to terms with the same base, which in this case is \(x\). By understanding and using the product of powers, complex multiplications with exponents become much simpler.