Problem 98 - Algebra A Combined Function

Question

Simplify each exponential expression. $$ \left(-14 a^{5} b c^{2}\right)\left(2 a b c^{4}\right) $$

Step-by-Step Solution

Answer

\(-28 a^6 b^2 c^6\)

Step 1: Distribute the Constants

To begin simplifying the expression, multiply the constants outside of the variables. Here, the constants are -14 and 2. Multiply these constants: \[-14 \times 2 = -28\]

Step 2: Apply the Product of Powers

Next, simplify the expression by applying the product of powers rule, which states that when you multiply like bases, you add their exponents. Let's do this for each variable:- For \(a\), add the exponents: \[a^5 \times a^1 = a^{(5+1)} = a^6\] - For \(b\), add the exponents: \[b^1 \times b^1 = b^{(1+1)} = b^2\] - For \(c\), add the exponents: \[c^2 \times c^4 = c^{(2+4)} = c^6\]

Step 3: Combine the Simplified Elements

Now, combine all the terms from the previous steps into one expression:\[(-28) \times a^6 \times b^2 \times c^6\]This can be written as:\[-28 a^6 b^2 c^6\]

Key Concepts

Product of Powers

The product of powers is a fundamental rule in simplifying exponential expressions. It tells us that when you multiply like bases, you simply add their exponents. This can be a great time-saver in algebra, allowing you to transform complicated multiplication into a simple addition.
For example, if you have \(a^m \times a^n\), the product of powers rule states that:

• The base, \(a\), stays the same.
• Add the exponents: \(m + n\).

So, \(a^m \times a^n = a^{m+n}\). This procedure simplifies expressions and reduces the clutter of multiplication.

Simplifying Expressions

Simplifying expressions means breaking them down into a more manageable or reduced form. This often involves combining like terms, using algebraic rules such as the product of powers, and performing arithmetic operations such as multiplication. 

• First, handle the constants (numerical coefficients) by multiplying or dividing as needed.
• Next, apply the product of powers to combine terms with the same base.
• Finally, ensure that all terms are presented in their simplest form for ease of interpretation and further operations.

In our exercise, we've simplified \( (-14)\) and \(2\) to \(-28\), and combined the exponents for \(a\), \(b\), and \(c\) based on the product of powers.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number or variable. They play a pivotal role in algebra by simplifying larger expressions.
To understand exponents, realize that:

• An exponent shows how many times the base is used as a factor. For example, \(a^3\) equals \(a \times a \times a\).
• Zero exponents mean \(a^0\) equals \(1\) for any non-zero \(a\).
• Negative exponents signify division, such as \(a^{-n} = \frac{1}{a^n}\).

In the product of powers, exponents simplify multiplying identical bases, allowing easy operations like addition within these context-specific rules.

Algebraic Multiplication

Algebraic multiplication involves multiplying terms in algebra, which can often include variables, constants, and exponents. Understanding this process is crucial for simplifying complex algebraic expressions.
Here are steps and tips to make it simpler:

• When you multiply two terms, multiply their coefficients (numbers) first, just like \(-14\) and \(2\) multiply to \(-28\).
• Next, apply the product of powers for any variables that appear more than once with exponents, as shown in \(a^5 \times a\) to get \(a^6\).
• Ensure that the final expression is in the simplest form, combining all similar terms.

With practice, algebraic multiplication becomes a straightforward and powerful tool in simplifying expressions and carrying out larger algebraic operations.