Problem 32 - Prealgebra

Question

The number \(\sqrt{54}\) lies between which two consecutive whole numbers?

Step-by-Step Solution

Answer

\(\sqrt{54}\) lies between 7 and 8.

Step 1: Identify Close Perfect Squares

To determine between which two whole numbers the square root of 54 lies, we first identify the perfect squares closest to 54. These are 49 (which is 7 squared) and 64 (which is 8 squared).

Step 2: Estimate Square Roots

Knowing that \(49 = 7^2 \)and \(64 = 8^2 \), we can conclude that \(7 < \sqrt{54} < 8 \).

Step 3: Verify Range

Verify that 54 is indeed between 49 and 64. Since 54 is greater than 49 and less than 64, it confirms that \(\sqrt{54}\) must lie between 7 and 8.

Key Concepts

Perfect Squares

Perfect squares are numbers that result from multiplying a whole number by itself. For example, when we multiply 7 by itself, we get 49. Therefore, 49 is a perfect square. It's important to recognize perfect squares because they are useful in various mathematical problems, including finding square roots and understanding the structure of numbers.

• Examples of perfect squares include: 1 (since 1x1=1), 4 (since 2x2=4), and 9 (since 3x3=9).
• Why perfect squares matter: Knowing perfect squares helps in problems like determining which two whole numbers a square root falls between.

It's essentially observing patterns and applying them to solve math exercises quickly.

Estimating Square Roots

Estimating square roots is a handy skill that helps you quickly approximate the value of a square root when it is not a perfect square. For the square root of 54, we need to see which perfect squares lie directly around it. Using the perfect squares closest to 54, which are 49 and 64, makes this easier.
Since we know:

• 49 is \(7^2\)
• 64 is \(8^2\)

We can understand that 54 falls right between these two, making the square root of 54 lie between 7 and 8. Estimating the square root doesn't give exact numbers but positions it correctly between whole numbers.
Practicing estimating square roots with other numbers and their closest perfect squares can strengthen your estimation skills.

Prealgebra Concepts

Prealgebra serves as the foundation for high school algebra and includes various mathematical concepts like order of operations, factors and multiples, and basic operations. One of the critical topics is understanding how to manipulate numbers and their properties, such as square numbers and square roots.

• Essential concepts in prealgebra include:
  • Understanding positive and negative numbers
  • Learning about factors, multiples, and divisibility
  • Grasping basic geometric shapes and measures
  • Working with fractions, decimals, and percentages
• Square roots often come into play in solving prealgebra problems, as they are a stepping stone to learning more complex algebraic equations.

Learning these basics is important as it not only helps in simple calculations but also builds a solid base for tackling more difficult math problems in the future.