Problem 76 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt{324}\)

Step-by-Step Solution

Answer

The square root of 324 is 18.

Step 1: Understanding the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. Here, we need to find the square root of 324, which means finding a number that when multiplied by itself results in 324.

Step 2: Estimating the Square Root

First, estimate the square root of 324 by identifying perfect squares close to 324. Know that 324 is between the perfect squares of 256 (since \(16^2 = 256\)) and 400 (since \(20^2 = 400\)). Since 324 is closer to 289 (\(17^2 = 289\)), it suggests that \(18^2\) might be a good estimate.

Step 3: Using Multiplication to Find the Exact Root

Multiply 18 by itself: \(18 \times 18 = 324\). Since this product matches the original number, 18 is confirmed as the square root of 324.

Step 4: Verify Using a Calculator

Double-check the calculation using a calculator to confirm: input \(\sqrt{324}\) into the calculator, which should return 18, verifying our manual calculation is correct.

Key Concepts

Perfect Squares

A perfect square is a special type of number that results from multiplying a whole number by itself. Understanding perfect squares is key to solving square root problems efficiently.
For instance, when you multiply 16 by itself, you get 256, making 256 a perfect square. Here, 16 is called a "root" of 256.

• Perfect squares of smaller numbers are essential to memorize as they can simplify finding square roots. Examples include:
  • 1, because \(1 \times 1 = 1\)
  • 4, because \(2 \times 2 = 4\)
  • 9, because \(3 \times 3 = 9\)

Memorizing these can give you a reference point when estimating or solving problems involving square roots. Recognizing perfect squares that the number falls between aids in efficiently estimating the root.

Estimate Square Root

Estimating the square root involves finding the closest perfect squares that encircle the given number. This narrows down potential candidates for the square root.
In the case of \(\sqrt{324}\), you begin by identifying nearby perfect squares, such as 289 and 400. Since 324 is closer to 289, noting that \(17^2 = 289\) and \(18^2 = 324\), suggests that 18 might be the square root.
This step is crucial because it allows you to use logical reasoning before confirming exact calculations. When the number you're working with isn't a perfect square, estimating can guide your calculation process.

Verify with Calculator

Once you've estimated and calculated what you believe is the square root, it's always a good practice to confirm the result using a calculator. This ensures accuracy and builds confidence in your multiplication attempt.
With a calculator, simply enter the square root function followed by your number. For \(\sqrt{324}\), the calculator will return 18, confirming your estimation and manual calculation.
This technique is especially important in more complex problems or when precision is crucial. Using a calculator helps verify manual work and serves as a useful tool for double-checking math assignments. Laying this foundation makes you proficient in dealing with square roots both manually and with technology.