Question

Which is the same as \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 ?\) A. \(9 \cdot 5\) B. \(5 \cdot 9\) C. \(5^9\) D. \(9^5\)

Short answer

C. \(5^9\)

Step-by-step solution

- Understand the question

The question asks for a more compact way to represent the expression given by multiplying nine 5's together: \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\).

- Recognize the pattern

When the same number is multiplied by itself multiple times, it can be written as an exponent. In this case, the number 5 is multiplied by itself 9 times.

- Write the expression using an exponent

We can write the expression as \(5^9\), where 5 is the base and 9 is the exponent. This notation means that 5 is used as a factor 9 times.

- Compare with the given multiple choice answers

The question provides the following options:A. \(9 \cdot 5\)B. \(5 \cdot 9\)C. \(5^9\) D. \(9^5\). From these options, we see that \(5^9\) matches our expression.

Key concepts

Multiplication

When we talk about multiplication, we are referring to the process of combining equal groups. For instance, if you have 3 groups of 4 apples each, you can calculate the total number of apples by multiplying 3 by 4. In this case, multiplication serves as a shortcut to addition.

For the original exercise, we multiplied 5 by itself multiple times: \[5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\].
When you encounter a situation where the same number is multiplied repeatedly, you can represent it using exponents to simplify it. This leads us to our next topic.

Base and Exponent Notation

Base and exponent notation allows us to write multiplication of the same number in a more compact form. In this notation:

• The base is the number being multiplied.
• The exponent shows how many times the base is used as a factor.

For example, \(5^3\) means 5 is multiplied by itself 3 times: 5 \cdot 5 \cdot 5.

In the exercise given, 5 is multiplied by itself 9 times, so we express this as \(5^9\). The number 5 is the base, and 9 is the exponent. This notation makes it easier to read and understand large multiplication expressions.

Simplifying Expressions

Simplifying expressions is a key concept in mathematics. It involves reducing a complex expression to a simpler or more compact form, making it easier to work with. By using exponents, large multiplication problems can be significantly simplified.

Let's revisit the original problem: \(5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\).
Using the base and exponent notation, we simplify this to \(5^9\). This way, we replace a long string of numbers with a single notation, making calculations more manageable.
Remember, exponents are powerful tools for simplifying repeated multiplication expressions and can save you a lot of time and effort in mathematical calculations.