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Chapter 3: Problem 17

Which of the following is a squared number? $$ 15,27,43,72,125,169 $$

Short Answer

Expert verified
169

Step by step solution

01

Understand Squared Numbers

A squared number, also known as a perfect square, is an integer that can be expressed as the product of an integer with itself.
02

Check Each Number

We need to find the square root of each given number and see if the result is an integer (whole number).
03

Calculate Square Roots

Compute the square roots: - √15 ≈ 3.87 (not an integer) - √27 ≈ 5.20 (not an integer) - √43 ≈ 6.56 (not an integer) - √72 ≈ 8.49 (not an integer) - √125 ≈ 11.18 (not an integer) - √169 = 13 (is an integer)
04

Identify the Squared Number

Since only 169 has an integer square root, it is the only squared number in the list.

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Key Concepts

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

Perfect Squares
Perfect squares are essential concepts in mathematics, especially when dealing with algebra and number theory. Simply put, a perfect square is an integer that can be obtained by squaring another integer.
For example:
  • The number 4 is a perfect square because it is equal to 2 * 2.
  • Similarly, 9 is a perfect square because it equals 3 * 3.
These numbers are called 'perfect' because they can be neatly made by multiplying an integer by itself.
Recognizing perfect squares is a handy skill. It helps in simplifying roots and understanding properties of numbers.
In our exercise, the number 169 is identified as the only perfect square.
Integer
An integer is any whole number, which can be positive, negative, or zero.
Some examples of integers are -3, 0, and 7.
An essential property of integers is that they do not have fractional or decimal parts.
When checking if a number is a perfect square, one must find the square root of that number and see if the result is an integer.
For instance:
  • The square root of 16 is 4. Since 4 is an integer, 16 is a perfect square.
  • Conversely, the square root of 15 is approximately 3.87, which is not an integer, hence 15 is not a perfect square.
This is why, in our given exercise, only 169 is considered a perfect square because its square root is an integer.
Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number.
For example, the square root of 25 is 5 because 5 * 5 = 25.
When finding square roots, we use the symbol √.
Calculating square roots can help you determine if a number is a perfect square.
If the result of the square root is an integer, the original number is a perfect square.
For example:
  • √36 = 6 (integer)
  • √50 ≈ 7.07 (not an integer)
Therefore, 36 is a perfect square, while 50 is not.
In the exercise provided, computations showed: `√15 ≈ 3.87`, `√27 ≈ 5.20`, `√43 ≈ 6.56`, `√72 ≈ 8.49`, `√125 ≈ 11.18`, and `√169 = 13`.
Only 169 produced an integer, confirming it as the perfect square.

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