Problem 69 - Algebra 1

Question

Write the expression in exponential form. \((a)(a)(a)(a)(a)(a)\)

Step-by-Step Solution

Answer

The expression \((a)(a)(a)(a)(a)(a)\) in exponential form is \(a^6\).

Step 1: Identify the base

Look at the term being repeated. This repeated term will become the base in the exponential form. In this case, the base 'a' is being multiplied.

Step 2: Count the repetitions

Count how many times the base 'a' is being multiplied by itself. In this case, 'a' appears 6 times.

Step 3: Write in exponential form

Now that we know the base 'a' and its count 6, we can write this in exponential form. The base goes first, then an exponent which is the count of how many times the base was repeated. This gives us \(a^6\).

Key Concepts

Base in Exponential Form

The base in exponential form plays a pivotal role in simplifying complex mathematical multiplications. In an expression such as \(a)(a)(a)(a)(a)(a)\), 'a' is the term that is being repeatedly multiplied. When we convert this repeated multiplication into exponential form, 'a' becomes the foundation, or the 'base', upon which the expression is built. Instead of writing 'a' multiple times, we can represent the multiplication of the same number or variable in a compact and efficient way using exponential notation.

In the given exercise, the base 'a' is identified as the term that appears repeatedly. The key to converting a repeated multiplication into exponential form is to discern which number or variable is being consistently multiplied by itself. Once this base is identified, the next step is to determine how frequently it appears. By understanding and correctly identifying the base, students can transform lengthy multiplication into a succinct and manageable exponential form, making their work both easier to handle and simpler to understand.

Exponent

An exponent is a powerful mathematical notation that conveys repeated multiplication of the same number. It is represented by a small number written above and to the right of the base. This small number is called the 'exponent'. In the context of the given exercise, after the base 'a' has been singled out, the next step is to count how many times 'a' repeats in the multiplication. The number of times the base is being multiplied by itself, which is 6 in our example, is then used as the exponent. Thus, the exponent in the expression \(a^6\) tells us that 'a' is being multiplied by itself six times.

By learning to identify and apply exponents correctly, students can drastically simplify mathematical expressions and calculations. For example, writing \(a^6\) rather than \(a)(a)(a)(a)(a)(a)\) saves space and reduces the potential for error in longer, more complex computations.

Repeated Multiplication

Repeated multiplication is the process of multiplying a number or variable by itself multiple times. It is a foundational concept in mathematics, especially when dealing with exponents. Taking our exercise as an example, the expression \(a)(a)(a)(a)(a)(a)\) involves multiplying 'a' by itself repeatedly, six times in total. This process can be tedious and space-consuming, especially when dealing with large numbers of repetitions.

Fortunately, understanding the concept of repeated multiplication leads to the recognition of patterns that can be more conveniently expressed in exponential form. Transitioning from repeated multiplication to exponential notation simplifies expressions, making them easier to work with and understand. Encouraging students to look for and leverage patterns in mathematics nurtures efficient problem-solving skills and can be a springboard to mastering more advanced concepts.