Problem 20 - Elementary and Intermediate Algebra

Question

Write each expression in an equivalent form using an exponent. $$ -5 u(-5 u)(-5 u)(-5 u)(-5 u) $$

Step-by-Step Solution

Answer

The expression is equivalent to (-5u)^5.

Step 1: Identify the Base

Notice that the expression contains repeated factors of -5u. The base of our exponent will be -5u since it is the integer-term pair being multiplied multiple times.

Step 2: Count the Repetitions

As we write in terms of an exponent, we need to determine how many times the base term (-5u) is repeated. Here, it appears 5 times: 1 instance alone and then 4 in parentheses.

Step 3: Write Using an Exponent

When a number or expression is multiplied by itself repeatedly, it can be written as a power. Since -5u is repeated 5 times, the expression is (-5u)^5.

Step 4: Apply to Original Expression

Replace the repeated multiplication in the original expression with the exponent form previously determined. As such, the expression is written as (-5u)^5.

Key Concepts

Base of an Exponent

In mathematics, an exponent is a powerful tool that helps simplify calculations involving repeated multiplication. At the core of this concept is the "base of an exponent". The base is the number or expression that is repeatedly multiplied. It's important to clearly identify it to correctly convert a multiplication scenario into an exponential form.

In the given exercise, when looking at the product \(-5u(-5u)(-5u)(-5u)(-5u)\), we see that "-5u" is consistently multiplied within the expression. Therefore, "-5u" is identified as the base.

To clarify, the term "base" in the exponentiation process is what we would repeatedly multiply as specified by the power. Recognizing this base sets the foundation for expressing any multiplicative repetition more concisely.

Repeated Multiplication

The concept of repeated multiplication is central to understanding exponents. It occurs when we multiply the same number or expression several times. Instead of writing out the entire multiplication sequence, we can express it concisely using powers.

For example, in our problem, \(-5u\) is multiplied by itself multiple times. Written in its long form, it looks like \(-5u(-5u)(-5u)(-5u)(-5u)\). Such repetitive processes are tedious and can lead to errors if not managed effectively.

This is where exponents simplify things. By transforming a string of multiplication into exponential form, we save time and reduce complexity. Think of it as a shorthand notation for multiplication! This becomes quite handy, especially with lengthy or complex expressions.

Writing Expressions as Powers

Finally, the ultimate goal of using exponents is converting repeated multiplications into a more manageable form called "powers". Writing expressions as powers involves taking the identified base and determining how many times it is used in the multiplication.

In the previous example, once "-5u" was identified as the base, the task was to determine how many times it repeated. We counted 5 instances. Therefore, the product could be rewritten compactly as \((-5u)^5\).

This not only makes the expression easier to comprehend but also facilitates further algebraic operations. Expressions in exponential form often reveal a clearer pattern or symmetry. And that simplicity can be used advantageously in solving complex mathematical problems.