Problem 17 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(-8 c^{4}\right)\left(2 c^{5}\right)$$

Step-by-Step Solution

Answer

-16 c^9

Step 1: Identify the expression

We are given the expression: \[ (-8 c^{4})(2 c^{5}) \]

Step 2: Apply the multiplication rules

We can break down the expression into two parts: the numerical coefficients and the algebraic terms with exponents. For the numerical coefficients, we have: \[ (-8) (2) = -16 \] For the algebraic terms with exponents, we have: \[ c^{4} c^{5} \] Recall that when multiplying expressions with the same base and exponents, we add the exponents. Therefore, we have: \[ c^{4} c^{5} = c^{4+5} = c^9 \]

Step 3: Combine the numerical coefficients and the algebraic terms

Now we will combine the numerical coefficients and the algebraic terms to get our final simplified expression: \[ -16 \cdot c^9 \] Therefore, the simplified expression is: \[ -16 c^9 \]

Key Concepts

Simplifying Expressions

When you're faced with an expression like \((-8 c^{4})(2 c^{5})\), the goal is to simplify it to its most reduce form without any negative exponents. Simplification involves breaking down the entire expression into smaller, more manageable parts. It can be helpful to separate numerical coefficients from the variables. Then, you can address each part separately.

The process goes as follows:

• Identify numbers and variables: In our example, numbers are \(-8\) and \(2\), while variables are part of \(c^{4}\) and \(c^{5}\).
• First, focus on the coefficients (\(-8\) and \(2\)). Simplify by multiplying these numbers together.
• Next, let's address the variables by applying the rules of exponents to them.

Simplifying may seem tricky at first, but once you break it into smaller steps, it becomes straightforward.

Multiplication Rules

To simplify expressions using multiplication rules, you need to understand how to multiply coefficients and apply the rules for exponents separately.

First, address numerical coefficients. When multiplying them, simply multiply as you usually would with regular numbers:

• Here, multipy \(-8\) and \(2\) resulting in \(-16\).

Next, focus on the algebraic terms, specifically those with exponents:

• The rule is that for terms with the same base, add their exponents.
• In our expression, the variable \(c\) is the base. The two terms with exponents are \(c^4\) and \(c^5\) which, when multiplied, require you to add the exponents: \(4 + 5 = 9\).

This step-by-step approach ensures accurate simplification of any algebraic expression with multiplication involved.

Algebraic Expressions

An algebraic expression like \((-8 c^{4})(2 c^{5})\) consists of both coefficients and terms that include variables with exponents. These expressions are common in algebra and can seem complex at first but, with the right strategies, they become more approachable.

When dealing with them:

• Identify each part: Constants \(-8\) and \(2\) are the straightforward part, while terms like \(c^4\) and \(c^5\) require more attention due to the exponents.

Remember, algebra is often about combining like terms and ensuring that terms are properly added or multiplied, adhering to mathematical rules. Keeping these rules in mind makes simplifying such expressions possible.

This understanding of algebraic expressions is crucial as it forms the base for solving more complex problems in algebra.