Problem 86 - Algebra 2

Question

Evaluate the expression \(a \cdot b^{n-1}\) for the given values of \(a, b,\) and \(n\). $$ a=1, b=2, n=5 $$

Step-by-Step Solution

Answer

The value of the expression is 16.

Step 1: Write Down the Expression

The expression you are asked to evaluate is \(a \cdot b^{n-1}\). This means you will be multiplying \(a\) by \(b\) raised to the power of \(n-1\).

Step 2: Substitute Given Values

Substitute \(a = 1\), \(b = 2\), and \(n = 5\) into the expression: \[1 \cdot 2^{5-1}\].

Step 3: Simplify the Exponent

Calculate \(n-1\) by subtracting 1 from \(n\). Thus, \(n-1 = 5-1 = 4\). The expression becomes \(1 \cdot 2^4\).

Step 4: Calculate the Power

Compute \(2^4\). This means multiplying 2 by itself 4 times: \(2 \times 2 \times 2 \times 2 = 16\).

Step 5: Multiply the Result

Multiply the result from Step 4 (which is 16) by \(a\), which is 1: \(1 \cdot 16 = 16\).

Key Concepts

Evaluate Expressions

To evaluate expressions in mathematics means to find their value by carrying out the necessary calculations. It's like decoding a message using the numbers and operators provided. In our exercise, this involves a combination of multiplication and exponentiation. You start by analyzing the expression given:

• Identify the different parts of the expression. In this case, we have numbers and an exponentiation operation.
• Follow the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When you evaluate an expression like \( a \cdot b^{n-1} \), you're applying this order of operations to arrive at a final numerical value. The resulting operation should be performed step-by-step to avoid confusion and errors.

Substitution

Substitution is a technique used to replace variables with specific numbers or simpler expressions. This is essential when you're working with abstract expressions in algebra. In the expression \( a \cdot b^{n-1} \), we substitute given values for the variables:

• Replace \( a \) with 1.
• Replace \( b \) with 2.
• Replace \( n \) with 5.

After making these substitutions, the expression becomes \( 1 \cdot 2^{5-1} \). This simplifies our work, allowing us to carry forward the evaluation with specific numbers instead of variables. Substitution simplifies abstract algebraic expressions into numerical ones, making evaluation possible.

Power of a Number

The power of a number refers to how many times a number, known as the base, is multiplied by itself. This is represented as an exponent. In our exercise, the expression contains \( 2^{4} \), which means the base 2 is raised to the fourth power.

• Calculate \( n-1 \) to identify the exponent of 2. Here, 5 minus 1 equals 4.
• 2 raised to the power of 4 means multiplying 2 by itself three more times: \( 2 \times 2 \times 2 \times 2 \), resulting in 16.

Understanding powers is crucial in evaluating expressions involving exponents. It tells you how many times to multiply the base by itself, giving exponential expressions their meaning and numeric value.