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Chapter 9: Problem 7

Simplify. $$ \sqrt{361} $$

Short Answer

Expert verified
The square root of 361 is 19.

Step by step solution

01

Understanding the Problem

The goal is to simplify the square root of 361. This means we need to determine what number, when squared, equals 361.
02

Recognizing Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. We need to find out if 361 is a perfect square.
03

Finding the Square Root

Check from smaller integers to see which one multiplied by itself equals 361. Through calculation, we find that 19 times 19 equals 361.
04

Conclusion

Since 19 squared is 361, the square root of 361 is 19.

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Key Concepts

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

Perfect Squares
Understanding perfect squares is key to simplifying square roots.
A perfect square is an integer that can be formed by multiplying another integer by itself. For example:
  • 1 is a perfect square because 1 * 1 = 1.
  • 16 is a perfect square because 4 * 4 = 16.
  • 25 is a perfect square because 5 * 5 = 25.

Recognizing these perfect squares helps in quickly identifying the square root of a number. In our exercise, you need to determine if 361 is a perfect square.
Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number.
For any given number 'n', a square root is represented as \( \sqrt{n} \). For example:
  • The square root of 16 is 4, because 4 * 4 = 16.
  • The square root of 25 is 5, because 5 * 5 = 25.
  • The square root of 49 is 7, because 7 * 7 = 49.

In our task, simplifying \( \sqrt{361} \) means finding an integer which, when multiplied by itself, equals 361.
Integer Multiplication
Integer multiplication is fundamental when working with square roots and perfect squares.
The idea is to check successive integers to see if their product results in the given number. For instance:
  • 2 * 2 = 4
  • 3 * 3 = 9
  • 4 * 4 = 16

To simplify \( \sqrt{361} \), we start testing integers to see which one, when multiplied by itself, equals 361. This may involve trial and error until we find that 19 * 19 = 361.
Mathematical Calculations
Mathematical calculations involve a variety of techniques to simplify problems.
When simplifying square roots, performing multiplication checks to verify perfect squares is a common method. In our example, we follow these steps:
  • Identify if 361 is a perfect square.
  • Perform integer multiplication to confirm which number squared gives 361.

Through successive multiplication and verification, we find that 19 multiplied by itself equals 361, thus \( \sqrt{361} = 19 \).
This approach can be applied to other numbers, helping simplify square roots efficiently.

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