Problem 60 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt{9}\)

Step-by-Step Solution

Answer

The roots of \(\sqrt{9}\) are \(3\) and \(-3\).

Step 1: Understand the Notation

The problem asks us to find the roots using our knowledge of multiplication. In this case, we are dealing with a square root function, indicated by the symbol \(\sqrt{}\). This means we need to find a number that, when multiplied by itself (squared), gives us the number inside the root (in this case, 9).

Step 2: Find the Possible Roots

We are looking for a number that, when squared, equals 9. We know from multiplication that \(3 \times 3 = 9\). However, we should also consider negative numbers because \((-3) \times (-3) = 9\) as well.

Step 3: Verify Each Root Using a Calculator

Use a calculator to check the results. When you enter \(3^2\), the calculator should display 9, confirming \(3\) as a root. Similarly, when \((-3)^2\) is entered, the calculator should also display 9, confirming \(-3\) as a root.

Key Concepts

Multiplication

Multiplication is a fundamental math operation where we combine equal groups to get a total. Often symbolized by '×', it involves adding a number to itself a specified number of times. When dealing with square roots, we employ multiplication to find numbers that result in a given product when multiplied by themselves.
For instance, to solve for the square root of 9 (\(\sqrt{9}\)), we ask: "What number times itself gives us 9?"

• We find that multiplying 3 by 3, stated as \(3 \times 3\), equals 9.
• This means 3 is one possible square root.

But it's crucial to remember that numbers can have more than one square root. Negative numbers can also be involved in multiplication, leading us to the additional root that we'll discuss next.

Negative Numbers

Negative numbers are simply numbers with a negative sign, indicating they are less than zero. When these numbers are multiplied together, particularly two negatives, the result is a positive product.
This concept ties directly to our square root problem since \(-3\times -3 = 9\), demonstrating another valid square root of 9.

• Although negative numbers might seem complex, always remember:
• Multiplying two negatives yields a positive.

This understanding helps in broadening the definition of square roots beyond just positive numbers and ensures we consider all possible solutions.

Calculator Verification

Calculator verification is an essential step in ensuring your manual calculations are correct, especially in a learning environment where mistakes can happen.
Using a calculator, you can quickly check the validity of your assumed square roots.

• Enter \(3^2\) into the calculator to verify that it returns 9.
• Repeat with \((-3)^2\) to observe that it also gives 9, confirming both calculations.

This process provides confidence in your understanding and ensures your answers are reliable. Additionally, getting accustomed to calculator functions can become a helpful tool for complex problems you encounter in the future.