Problem 57 - Algebra 2

Question

Writing Is 10 a first root of 10\(?\) Explain.

Step-by-Step Solution

Answer

No, 10 is not a first root of 10, because \(10 \times 10 = 100\), not 10.

Step 1: Understand the concept of root

A root of a number is a value that, when multiplied by itself a certain number of times, equals the original number. In this case, we're talking about the first root (also known as the square root), which means the number needs to be multiplied by itself once to get the original number.

Step 2: Verify if 10 is a first root of 10

To find out if 10 is a first root of 10, we need to multiply 10 by itself. If the result is the same as the original number, then 10 is indeed a first root of 10.

Step 3: Perform the operation

Multiplying 10 by itself, we get \(10 \times 10 = 100\). This is not equal to the original number, which is 10.

Step 4: Draw the conclusion

Since the result of the multiplication is not the original number, we can conclude that 10 is not a first root of 10.

Key Concepts

Roots of Numbers

Finding the roots of numbers is an important concept in mathematics. A root is basically a number that, when multiplied by itself a certain number of times, gives the original number. There are different types of roots based on how many times the number is multiplied by itself, like square roots or cube roots.

• The **square root** of a number is what you multiply by itself once to get that number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\).
• The **cube root** of a number is what you multiply by itself two more times to get that number. For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\).

In the exercise, we were asked if 10 is the square root or "first root" of 10. However, when checking this, multiplying 10 by itself does not return 10, so 10 is not the square root of itself.

Multiplication

Multiplication is a fundamental mathematical operation that simplifies the process of adding a series of identical numbers. When you multiply, you're essentially adding a number to itself a certain number of times. This operation is represented by the symbol \( \times \) or sometimes as simply a dot.

• For example, \( 4 \times 3 \) means you add the number 4 to itself three times, resulting in 12.
• Another example, \( 7 \times 2 \), is adding the number 7 to itself once, resulting in 14.

When we applied multiplication in the exercise to test if 10 is its own square root, we multiplied 10 by itself: \( 10 \times 10 = 100 \). The result did not match the original number, proving 10 cannot be its own square root.

Mathematical Operations

Mathematical operations are processes that help us solve equations and understand relationships between numbers. These include addition, subtraction, multiplication, and division. Recognizing which operation to use is critical in solving math problems effectively.

• **Addition** helps combine numbers, increasing their cumulative value.
• **Subtraction** helps find the difference between numbers, reducing the original value.
• **Multiplication** aids in simplifying repeated addition.
• **Division** helps in distributing a number into equal parts.

In the context of the exercise, multiplication was used to test if 10 could be its own square root. Understanding how to use these operations correctly is key to navigating through mathematical challenges efficiently.