Problem 66 - Algebra 1: Concepts and Skills

Question

Write the expression in exponential form.(Lesson 1.2) $$ 2 b \cdot 2 b \cdot 2 b $$

Step-by-Step Solution

Answer

The exponential form of the expression is \((2b)^3\).

Step 1: Identify the base and number of times it appears

In the given expression, our base is \(2b\) and it appears three times. It means that \(2b\) is multiplied by itself three times.

Step 2: Write the expression in exponential form

Given that our base is \(2b\) and that it is used as a factor three times, we express this as \((2b)^3\).

Key Concepts

Writing Expressions in Exponential Form

In algebra, it's important to simplify expressions so they're easier to work with. One way to do this is by writing them in exponential form. When an expression involves a number or variable being multiplied by itself multiple times, we can represent it more concisely with exponents.

For instance, consider the multiplication of the same base, such as in the expression 2b \times 2b \times 2b. We can see that the base, 2b, is used as a factor three times. To write the expression in exponential form, we raise the base to the power of the number of times it appears. This procedure gives us a new expression, \( (2b)^3 \).

• Identify the base: Look for the factor that repeats through multiplication.
• Count the number of times the base is multiplied by itself.
• Write the base followed by an exponent, which is the count of repetitions.

Using this strategy makes expressions simpler to understand and helps students save time when performing calculations.

Exponential Notation

Exponential notation is a shorthand way to express repeated multiplication of the same factor. It consists of two parts: the base and the exponent. The base is the value that is being multiplied, and the exponent, written as a small number above and to the right of the base, tells us how many times to multiply the base by itself.

For example, \( (2b)^3 \) is an exponential notation where \(2b\) is the base and \(3\) is the exponent. This tells us to multiply \(2b\) by itself three times. In general, if \( a \) is the base and \( n \) is the exponent, the exponential notation \( a^n \) means \( a \) is multiplied by itself \( n \) times. Understanding this notation is fundamental in algebra as it simplifies complex operations and is used extensively in equations and formulas.

Identifying the Base in Expressions

The base is a critical component in understanding exponential expressions. It's the factor that is raised to a power. To identify the base in an expression, look for the common factor in a sequence of multiplications.

In the expression \( (2b)^3 \), 2b is the base. Recognizing the base is essential before applying the exponent, because it tells us which number or variable is getting multiplied.

Common Challenges in Identifying the Base

• Do not confuse coefficients (numerical factors) with bases; the base can include both a number and a variable.
• In expressions with multiple variables or numbers, group the repeated factors to find the correct base.
• Always check the entire expression to ensure you aren't missing any factors included in the base.

Correctly identifying the base ensures accurate simplification of algebraic expressions and is the first step towards mastering exponential notation.