Problem 13 - Basic Engineering Mathematics

Question

Simplify \(a^{3} b^{2} c \times a b^{3} c^{5}\)

Step-by-Step Solution

Answer

The simplified expression is \(a^4 b^5 c^6\).

Step 1: Identify Like Terms

In the expression \(a^3 b^2 c \times a b^3 c^5\), identify the like terms. We can see there are common base variables: \(a\), \(b\), and \(c\).

Step 2: Apply the Product Rule for Exponents

The product rule states that when multiplying like bases, you add the exponents. Apply this to each variable: \(a^{3+1} = a^4\), \(b^{2+3} = b^5\), and \(c^{1+5} = c^6\).

Step 3: Write the Simplified Expression

Combine the results from the previous step to write the expression: \(a^4 b^5 c^6\).

Key Concepts

Exponents

Exponents represent repeated multiplication of a base number. For example, in the term \(a^3\), \(a\) is the base and 3 is the exponent, indicating that \(a\) is multiplied by itself three times: \(a \times a \times a\). Exponents make it easier to work with large numbers and simplify expressions.
When working with exponents, it's important to remember:

• "Power" refers to the exponent.
• The "base" is the number multiplied by itself.
• Raising a number to an exponent of 0 always results in 1, regardless of the base (except if the base is 0).
• Multiplying exponents with the same base follows specific rules such as the product rule.

Understanding exponents simplifies mathematical expressions and helps in both basic and advanced algebra problems. This concept is crucial when manipulating algebraic expressions like \(a^3 b^2 c\) in our example.

Product Rule

The product rule for exponents is a fundamental rule in algebra. It simplifies the multiplication of terms that have the same base by adding their exponents.
Here's how it works:

• When multiplying terms like \(a^m\) and \(a^n\), the exponents are added: \(a^{m+n}\).
• This rule applies to each variable or number with the same base.

Looking at the expression \(a^3 b^2 c \times a b^3 c^5\), we apply the product rule to each set of like bases:

• For \(a\): \(a^{3+1} = a^4\)
• For \(b\): \(b^{2+3} = b^5\)
• For \(c\): \(c^{1+5} = c^6\)

Using the product rule simplifies the process of combining powers and makes solving algebraic equations quicker and more efficient. This is particularly useful in problems with multiple terms and variables.

Like Terms

"Like terms" are terms in an algebraic expression that have identical variable parts, even if their coefficients differ. Recognizing like terms is vital as it allows you to combine and simplify expressions efficiently.
For example:

• In the expression \(3a + 2a\), both terms have the variable \(a\) making them like terms, and they can be combined: \(5a\).
• "Like" does not mean identical; \(4b^2\) and \(7b^2\) are like terms because they share the same variable part \(b^2\).

In our original expression \(a^3 b^2 c \times a b^3 c^5\), each group of like terms \(a^m, b^n,\) and \(c^o\) is identified and combined using the product rule.
Identifying like terms is crucial for simplifying expressions and making them more manageable. It can drastically reduce the complexity of solving algebraic problems and ensure you achieve the simplest form possible.