Problem 13 - Elementary Algebra

Question

\(\left(9 x^{-3}\right)\left(4 x^{5}\right)\)

Step-by-Step Solution

Answer

36x^{2}

Step 1 - Identify the expression components

Look at the expression \( \left(9 x^{-3}\right)\left(4 x^{5}\right) \ \). There are two main parts: a coefficient and a variable with an exponent in each term.

Step 2 - Multiply the coefficients

Multiply the coefficients of each term: \( 9 \text{ and } \ 4 \). The product is \ 9 \times 4 = 36.\

Step 3 - Add the exponents

Apply the property of exponents that states \ x^a \cdot x^b = x^{a+b} \ \. Add the exponents of \( x \) in the expression \( x^{-3} \ and \ x^{5} \). The sum of the exponents is \ -3 + 5 = 2 \.

Step 4 - Combine the results

Combine the product of the coefficients and the sum of the exponents to rewrite the expression. Thus, \( 36 x^{2} \ \).

Key Concepts

Coefficients

A coefficient is the numerical part of a term that includes a variable. For example, in the expression \(9x^{-3}\), 9 is the coefficient as it represents how many units of \(x^{-3}\) are included. Identifying coefficients is crucial when multiplying expressions since it is one of the first steps. Recall that when multiplying two terms, you first multiply the coefficients. In the exercise, the coefficients 9 and 4 were identified, and their product was calculated as follows:
\(9 \times 4 = 36\).
By separating numerical parts (coefficients) from the variables, you simplify the multiplication process.

Exponents

An exponent tells you how many times a base (like \(x\) in our exercise) is multiplied by itself. For instance, in \(x^{-3}\), -3 is the exponent and indicates reciprocal action. Negative exponents, like in our example \(x^{-3}\), can be tricky but follow consistent rules. When you have \(x^{-a}\), it translates to the reciprocal of \(x^a\). In our case, we consider exponents while multiplying because they follow specific properties that make the simplification process smoother.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. In the exercise, this involves both coefficients and exponents in the following steps:

• Multiply coefficients: From \(9x^{-3}\) and \(4x^5\), we got \(9 \times 4 = 36\).
• Add exponents: Using the rule - \(x^a \times x^b = x^{a+b}\) - combining\(x^{-3}\) and \(x^5\) gives \(x^{-3+5} = x^2\).

Each part gets simplified separately and then combined into a final, more manageable expression. Thus, \(36x^2\) is the simplified form.

Properties of Exponents

When dealing with exponents, certain rules make the calculations easy and consistent. For multiplication, the primary property is:
\[x^a \times x^b = x^{a+b}\].
This means you add the exponents of the same base. This property was applied in our exercise to combine \(x^{-3}\) and \(x^5\). Understanding this rule ensures you handle various exponents scenarios correctly. Subtracting the negative exponent from a positive one, as done here, yields the straightforward \(x^2\). Using the properties of exponents correctly simplifies complex expressions efficiently.