Problem 18 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(3 w^{9}\right)(-7 w)$$

Step-by-Step Solution

Answer

The simplified expression is \(-21 w^{10}\).

Step 1: Multiply constant terms.

First we need to multiply the constant terms which are 3 and -7. \(3 \times (-7) = -21\).

Step 2: Multiply variable terms.

Next, we need to multiply variable terms which are \(w^{9}\) and \(w\). To do this, we need to add the exponents, according to the rules of indices: \(w^{a} \times w^{b} = w^{a+b}\). Thus, we have \(w^{9} \times w = w^{9+1}\).

Step 3: Simplify the exponents.

Simplify the exponent in the last step: \(w^{9+1} = w^{10}\).

Step 4: Combine constant and variable terms.

Now, we can combine the constant term from Step 1 and the variable term from Step 3: \(-21 \cdot w^{10}\). The simplified expression is \(-21 w^{10}\).

Key Concepts

Negative Exponents

Negative exponents might seem a bit scary at first, but they are just another way of expressing fractions or reciprocals. A negative exponent means you divide, instead of multiply. For instance, \( x^{-3} \) is the same as \( \frac{1}{x^3} \). This means that negative exponents flip the base number to the other side of the fraction line.

• \( 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \)
• \( y^{-1} = \frac{1}{y} \)

It's important to remember that expressions should not remain with negative exponents. You simplify them to positive ones, like shown in the above examples, to make your expressions easier to understand and work with. In this exercise, we successfully kept our exponents positive throughout.

Multiplication of Variables

When multiplying variables, the key rule is to look at their exponents. The variable with itself gets multiplied, and then you add their exponents. Think of exponents as a count of how many times a number is used in a multiplication. For variables, this translates into:

• If you are multiplying \( w^3 \times w^4 \), you will add the exponents: \( 3 + 4 = 7 \), resulting in \( w^7 \).

This is because \( w^3 \) means \( w \cdot w \cdot w \), and \( w^4 \) means \( w \cdot w \cdot w \cdot w \). So altogether, you have \( w \) multiplied by itself seven times. In the exercise, we multiplied \( w^9 \) with \( w \), adding the exponents to get \( w^{10} \). This ensures the expression accurately reflects how the variables interact.

Rules of Indices

The rules of indices, or laws of exponents, are essential guidelines to help you handle powers in algebraic expressions. Here are the basic ones:

• Product Rule: When multiplying like bases, you add the exponents: \( x^a \times x^b = x^{a+b} \).
• Quotient Rule: When dividing like bases, you subtract the exponents: \( \frac{x^a}{x^b} = x^{a-b} \).
• Power of a Power Rule: When taking a power of a power, you multiply the exponents: \((x^a)^b = x^{a\cdot b}\).

In this exercise, we utilized the product rule. This was shown when we combined \( w^9 \) and \( w \) to form \( w^{10} \). These rules help simplify expressions and make calculations easier. Mastering them is a fundamental step in excelling at algebra and simplifying expressions efficiently.