Problem 62 - Elementary Algebra Within Reach

Question

In Exercises \(59-62\), rewrite the expression as a product. $$ (3.5)^{8} $$

Step-by-Step Solution

Answer

Therefore, \((3.5)^{8}\) can be written as a product as \(3.5 * 3.5 * 3.5 * 3.5 * 3.5 * 3.5 * 3.5 * 3.5\).

Step 1: Recognizing the Expression

The given expression is \((3.5)^{8}\), which is the same as saying 3.5 raised to the power of 8. This can be expressed as a multiplication.

Step 2: Expressing as a Product

The expression can be written in product form as \(3.5 * 3.5 * 3.5 * 3.5 * 3.5 * 3.5 * 3.5 * 3.5\), which is eight times multiplication of 3.5.

Key Concepts

Expression Simplification

Simplifying algebraic expressions is an essential skill in algebra that can make complex problems more manageable. Expression simplification involves reducing expressions into their simplest form by combining like terms, eliminating unnecessary elements, or by rewriting them in a more concise manner.

For instance, when dealing with exponentiation, such as with the expression \(3.5^{8}\), simplification doesn't inherently change the value but rather changes the manner in which it is presented. Simplifying an exponential expression involves understanding the properties of exponents and recognizing patterns. In the exercise provided, simplification would not further reduce the expression's complexity, as it is already a simple exponential form. However, expressing it as a product, as shown in the step-by-step solution, is a form of rewriting that can be considered a way to simplify depending on context.

Writing Expressions as Products

The process of writing expressions as products serves a vital function in algebra. It translates exponential terms into repeated multiplications, which can sometimes provide more insight or a different perspective on the problem at hand.

When it comes to the problem \(3.5^{8}\), writing the expression as a product means to present it as \(3.5 \times 3.5 \times 3.5 \times 3.5 \times 3.5 \times 3.5 \times 3.5 \times 3.5\). This method breaks down the exponentiation into an extended multiplication, making it clear that 3.5 is used as a factor exactly eight times. Such a representation can be particularly useful when the exponentiation is more complex or when visualizing the impact of each multiplication on the overall value.

Powers and Exponents

Powers and exponents are fundamental concepts in algebra that represent repeated multiplication. An exponent, usually a small number written above and to the right of a base number, indicates how many times that base should be used as a factor.

The expression \(3.5^{8}\) consists of a base, 3.5, and an exponent, 8, which tells us that 3.5 should be multiplied by itself a total of eight times. Effectively, powers are a shorthand notation to denote large-scale multiplications. Understanding how to work with exponents is crucial for performing operations with expressions and simplifying accordingly. Key properties of exponents, such as product, quotient, and power rules, can tremendously expedite the simplification process when dealing with powers in algebraic expressions.