Problem 28 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. $$ (9 a)(9 a)(9 a)(9 a) $$

Step-by-Step Solution

Answer

Question: Convert the expression (9a)(9a)(9a)(9a) into exponential notation. Answer: The expression in exponential notation is 6561a^4.

Step 1: Identify the base and the exponent

In the given expression, 9 and a are the bases while the exponent is 4 because each of the bases gets multiplied by themselves 4 times.

Step 2: Rewrite the expression in exponential notation

Use the properties of exponents to rewrite the expression in exponential notation, following the rule \((ab)^n = a^n b^n\). Here, the base is 9a, and the exponent is 4. Thus, we get: $$ (9a)(9a)(9a)(9a) = (9a)^4. $$

Step 3: Apply the exponent to each factor

Next, apply the exponent to each factor within the base (i.e., '9' and 'a'). This is accomplished using the same rule \((ab)^n = a^n b^n\): $$ (9a)^4 = 9^4 a^4. $$

Step 4: Simplify the expression

Compute 9^4 to simplify the expression: $$ 9^4 a^4 = 6561a^4. $$ Now, the expression is written in exponential notation: $$ (9 a)(9 a)(9 a)(9 a) = 6561a^4. $$

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when working with exponential notation. Exponents indicate how many times a base is multiplied by itself. For instance, in the expression \( 9^4 \), the number 9 is the base, and 4 is the exponent, which tells us that 9 is multiplied by itself 4 times. There are several key rules for working with exponents.

Multiplication Rule

The multiplication rule for exponents states that when you multiply two powers with the same base, you add the exponents. This is expressed as \( a^n \times a^m = a^{n+m} \).

Power Rule

The power rule explains that to raise a power to another power, you multiply the exponents. So for \( (a^n)^m \), the result is \( a^{n \times m} \).

Distributive Rule of Exponents

This rule, also known as the distributive property of exponents over multiplication, shows us how to handle expressions like \( (ab)^n \), which is equivalent to \( a^n \times b^n \). In the problem we're discussing, we applied this rule to rewrite \( (9a)^4 \) as \( 9^4 a^4 \).

By familiarizing themselves with these properties, students can solve exponential notation problems more effectively and understand why certain steps are taken in the simplification process.

Simplifying Expressions

Simplifying mathematical expressions involves rewriting them in a more concise and efficient way without changing their value. The goal is to make the expression easier to understand and work with. When dealing with exponential expressions, simplification typically includes applying the properties of exponents to combine like terms, convert to exponential notation, and reduce expressions to their smallest possible form.

For example, computing the value of the exponent, like \( 9^4 \), simplifies the expression from a repeated multiplication to a single number, 6561 in this case. This process is crucial because it reduces the complexity of calculations and aids in the understanding of the underlying structure of the expression. In educational terms, simplifying expressions serves not only to provide a shorter answer but also to showcase the fundamental principles that govern arithmetic and algebraic operations.

Exponential Expressions

Exponential expressions, such as \( 9^4 a^4 \), contain exponents and represent repeated multiplication. These expressions can be daunting for students, but understanding how to construct and deconstruct them is vital for mastering algebra and higher-level mathematics.

When we break down \( 9^4 a^4 \) into \( (9a)(9a)(9a)(9a) \), it becomes clear that the exponential notation is a shorthand for this longer multiplication process. Delving into exercises that require conversion between expanded and exponential form helps reinforce the concept that exponential notation is merely a more efficient way to write out multiplication involving the same factor. Furthermore, it emphasizes the importance of the base and exponent, and how they can quickly tell us how an expression can grow or shrink in value, which is especially useful in fields such as finance and science where exponential growth or decay are often analyzed.