Problem 29 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. $$ (-7)(-7)(-7) \text { aabbba }(-7) \text { baab } $$

Step-by-Step Solution

Answer

Question: Express the given expression in exponential notation: $(-7)(-7)(-7)(-7) \text{ aabbba }(-7) \text{ baab }$ Answer: $(-7)^4 \text { aabbba }(-7) \text { baab }$

Step 1: Count the number of repeated factors

Observe the expression given $$ (-7)(-7)(-7)(-7) $$ Notice that the number -7 is being multiplied with itself 4 times.

Step 2: Write the exponential notation for the repeated factors

Since the number -7 is multiplied with itself 4 times, we can write this as $$ (-7)^4 $$

Step 3: Combine the exponential notation with the given text

Now that we have the exponential notation for the numbers, we can insert the text back into the expression as it is just given in the original expression. The final expression is: $$ (-7)^4 \text { aabbba }(-7) \text { baab } $$

Key Concepts

Multiplication

In mathematics, multiplication is one of the fundamental operations. It is often considered as repeated addition. For example, if you have 4 baskets, each containing 3 apples, you could find the total number of apples by adding 3 four times: 3 + 3 + 3 + 3. This is the same as saying 4 times 3, or writing it as the multiplication expression 4 \(\times\) 3.

When we multiply numbers, we're essentially combining groups of equal sizes into a larger group. This becomes especially useful when dealing with large numbers or numerous repetitions of a smaller number. In our particular problem, we see that the number -7 is multiplied by itself repeatedly, which efficiently displays the power of multiplication in compressing repeated operations into a concise expression.

Repeated Factors

Repeated factors are core to understanding multiplication in exponential notation. When a number is multiplied by itself multiple times, each instance of the number is referred to as a 'factor'.

In the given problem, the number -7 appears as a repeated factor 4 times: \((-7) \times (-7) \times (-7) \times (-7)\). Seeing this directly showcases how repeated factors make up the basis of an exponent. The use of repeated factors streamlines mathematical expressions, making calculations easier to follow and less prone to error.

• Each repeated instance of the factor counts towards the overall exponent.
• It simplifies writing long expressions where the same number appears multiple times in a product.

These repeated factors are at the heart of creating a concise representation using exponents.

Exponent

The concept of an exponent is a shorthand way to denote how many times a number (the base) is multiplied by itself. It is written as a small number to the top right of the base number. In our problem, writing \((-7)^4\) is a clear example of utilizing an exponent. This reads as "negative seven to the fourth power".

Exponents help manage larger calculations, labeling the base and showing the number of times it repeats succinctly:

• The base is -7, representing the number being multiplied.
• The exponent is 4, indicating -7 is used four times as a factor.

Through exponents, we simplify complex multiplication processes. This reduces length and complexity, maintaining elegance and clarity in mathematical expressions.