Problem 2 - Elementary Algebra

Question

Write each of the following using exponents. \((3 b)(3 b)(5 c)(5 c)(5 c)(5 c)\)

Step-by-Step Solution

Answer

Question: Rewrite the expression \((3 b)(3 b)(5 c)(5 c)(5 c)(5 c)\) using exponents. Answer: \(3^2 b^2 \cdot 5^4 c^4\)

Step 1: Identify the bases and count their occurrences

In the given expression \((3 b)(3 b)(5 c)(5 c)(5 c)(5 c)\), there are two bases being multiplied repeatedly: 3 and 5. For the variable part, we have bases b and c. Now let's count their occurrences: - The base 3 occurs twice in the expression. - The base 5 occurs four times in the expression. - The base b occurs twice in the expression. - The base c occurs four times in the expression.

Step 2: Rewrite the expression using exponents

Now that we have identified the bases and counted their occurrences, we can rewrite the expression using exponents. For each base, write the base followed by a power equal to the number of times it occurs in the expression: - Rewrite \((3 b)(3 b)\) as \(3^2 b^2\). - Rewrite \((5 c)(5 c)(5 c)(5 c)\) as \(5^4 c^4\).

Step 3: Combine the expressions

Finally, we can combine the expressions from Step 2 to get the final answer: \(3^2 b^2 \cdot 5^4 c^4\).

Key Concepts

Bases

In mathematics, a base is a number or variable that is repeatedly multiplied by itself. Understanding bases is essential for dealing with exponents. In the given expression \((3 b)(3 b)(5 c)(5 c)(5 c)(5 c)\), we identify the bases as the parts of the expression that are raised to a power or exponent. For example:

• The base 3 in the expression indicates that we are multiplying the number 3 repeatedly.
• Similarly, the base 5 indicates that the number 5 is being multiplied by itself several times.
• For the variables, base \(b\) and base \(c\) are used in a similar way. They are also multiplied repeatedly.

Bases are key to understanding exponents, as they form the foundation from which exponential expressions build.

Occurrences

Counting occurrences is all about recognizing how many times each base appears in a multiplication sequence. This process is crucial when rewriting expressions using exponents. For the task at hand, we identified the number of times each base appears:

• Base 3 occurs twice.
• Base 5 occurs four times.
• Base \(b\) occurs twice.
• Base \(c\) occurs four times.

Each count tells us the power or exponent to which the base should be raised. By understanding occurrence, we can effectively convert products into exponents, simplifying our mathematical work.

Rewrite Using Exponents

Rewriting expressions using exponents is a way to simplify multiplicative expressions. It turns a long multiplication sequence into a more compact and readable form. With the counts from earlier, we can now express the same operation more succinctly:

• Instead of writing \((3 b)(3 b)\), we use the exponentiated form \(3^2 b^2\).
• For \((5 c)(5 c)(5 c)(5 c)\), we write \(5^4 c^4\).

Exponents reflect the number of times a base is multiplied by itself. This not only makes expressions neater but also facilitates further manipulation, especially in complex algebraic tasks. It is an important skill widely used in algebra and higher-level math.

Algebraic Expression Manipulation

Algebraic expression manipulation is the process of transforming and simplifying expressions to solve equations or to make calculations easier. After rewriting each segment of an expression with exponents, the next logical step is to combine these into a single expression. In this case:

• We combine \(3^2 b^2\) and \(5^4 c^4\) as products since they represent parts of the same original expression.

Thus, our final expression looks like \(3^2 b^2 \cdot 5^4 c^4\). Such manipulation of expressions is central to algebra, allowing complex problems to be tackled in a structured and simpler format. It enables efficient computation and clarity in mathematical reasoning.