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Chapter 9: Problem 8

Evaluate each radical without using a calculator or a table. (Objective 1) $$\sqrt{2500}$$

Short Answer

Expert verified
The square root of 2500 is 50.

Step by step solution

01

Understand the Radical Expression

The problem asks us to evaluate \( \sqrt{2500} \). This means we need to find the number that, when multiplied by itself, gives us 2500.
02

Estimate the Root

To evaluate \( \sqrt{2500} \), let's first estimate. We know that \( 50^2 = 2500 \) and that's a perfect square because multiplying 50 by 50 results in 2500.
03

Evaluate the Square Root

Since \( 50^2 = 2500 \), then \( \sqrt{2500} = 50 \). This checks out because 50 times 50 is indeed 2500.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Radical Expressions
A radical expression involves using the radical symbol \( \sqrt{} \), which represents the square root of a number. Square roots are special because they unravel or "root out" the number under the radical sign, which we call the radicand. In the expression \( \sqrt{2500} \), 2500 is the radicand, and the whole expression asks for the number which, when squared (or multiplied by itself), equals 2500.

To evaluate a radical expression, it's crucial to understand that finding the square root is essentially reversing the exponentiation process. If the result is a whole number, such as in this case, it is known as a perfect square, which leads us to our next topic.
Perfect Squares
Perfect squares are integers that result from squaring whole numbers. For example, numbers like 1, 4, 9, 16, and 2500 are perfect squares because they can be expressed as the square of integers 1, 2, 3, 4, and 50, respectively.

The process of identifying if a number like 2500 is a perfect square involves estimating whether there's a whole number whose square equals the original number. It's important to remember that:
  • Perfect squares can greatly simplify the evaluation of radical expressions because once you recognize a perfect square, determining the square root is straightforward.
  • Checking multiplication is a key strategy. For instance, \( 50^2 = 2500 \), which shows 2500 is a perfect square as \( 50 \times 50 = 2500 \).
In our exercise, knowing 2500 is a perfect square made evaluating \( \sqrt{2500} \) more efficient.
Estimation of Square Roots
Before solving for an exact square root, making an estimation can be incredibly helpful. Estimating involves finding two perfect squares that the given number falls between and then narrowing down the possibilities. For instance, to estimate \( \sqrt{2500} \):
  • Recognize nearby perfect squares. We know that \( 2500 \) is close to \( 2304\) (\(48^2\)) and \(2601\) (\(51^2\)).
  • Since 2500 is exactly \( 50^2 \), estimation confirms the root is a whole number, 50.
Estimation is a fundamental technique to give a ballpark figure,allowing for a guided approach to finding the actual square root, especially when a calculator or table is not available.

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