Problem 11 - Elementary Algebra

Question

\(b^{11} b^{4}\)

Step-by-Step Solution

Answer

b^{15}

Step 1: Identify the Base and Exponents

Notice the bases and the exponents in the given expression. Here, both terms have the same base, which is 'b'. The exponents are 11 and 4 respectively.

Step 2: Apply the Exponent Addition Rule

When multiplying terms with the same base, add the exponents. So, for the expression, it will be: b^{11+4}

Step 3: Simplify the Exponents

Add the exponents together: 11 + 4 = 15 Now the expression becomes: b^{15}

Key Concepts

Exponent Addition

Exponent rules are essential for simplifying and solving algebraic expressions that involve powers of numbers or variables. One critical rule is the **Exponent Addition Rule**. This rule states that when you multiply two exponents with the same base, you simply add the exponents together. For example, consider the expression given in the exercise:

• The first term is \(b^{11}\).
• The second term is \(b^{4}\).

Notice that both terms have the same base, which is 'b'. The exponents here are 11 and 4. To simplify the expression, you add the exponents:
\(b^{11+4} = b^{15}\). This is because of the Exponent Addition Rule, i.e., \(a^m \cdot a^n = a^{m+n}\), where 'a' is the base and 'm' and 'n' are the exponents.

Base in Algebra

Understanding the concept of the base in algebra is fundamental. The base is the number or variable that is being raised to a power or exponent. In the expression \(b^{11} \cdot b^{4}\), the base is 'b'. It's crucial to identify the base correctly because the rules for dealing with exponents depend on having the same base.

• If the bases are the same, you can apply exponent rules like addition or subtraction.
• If the bases are different, you cannot directly apply these rules and may need to simplify further or use other algebraic techniques.

In our exercise, since the base 'b' is the same for both terms, we can apply the Exponent Addition Rule to simplify the expression.

Simplifying Exponents

After identifying the base and applying the appropriate exponent rules, the next step is **Simplifying Exponents**. Simplification makes the expression easier to work with in further mathematical operations. In our example, we had the expression \(b^{11} \cdot b^{4}\).

• By applying the Exponent Addition Rule: \(b^{11+4} = b^{15}\).

Now, the complex multiplication of exponents is reduced to a single term with a higher exponent, \(b^{15}\). This not only makes the expression more manageable but also helps in understanding the underlying structure of the algebraic terms involved.