What is the square root of 69

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what is the square root of 69

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What is the square root of 69?

The square root of 69 is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal representation goes on forever without repeating. However, we can approximate it using a calculator as follows:

\sqrt{69} \approx 8.306623862918075

To provide more detail:

  1. Understanding Square Roots:

    • The square root of a number ( x ) is a number ( y ) such that ( y^2 = x ).
    • For 69, the square root is denoted as ( \sqrt{69} ).

  2. Approximation Methods:

    • Using a calculator will give ( \sqrt{69} \approx 8.30662 ) or more precisely ( 8.306623862918075 ).
    • This is useful for most practical purposes.

  3. Radical Form:

    • The square root of 69 in its simplest radical form remains ( \sqrt{69} ) since 69 is not a perfect square and does not factor into smaller square numbers.

  4. Estimation Approach:

    • To estimate without a calculator, note that ( 64 < 69 < 81 ).
    • Thus, \sqrt{64} = 8 and \sqrt{81} = 9 so ( \sqrt{69} ) is between 8 and 9.

  5. Use Cases:

    • Square roots are frequently used in various fields such as engineering, physics, and mathematics for solving quadratic equations, among other applications.

In conclusion, when you need the square root of 69, the approximate value is about 8.31, but the most accurate representation for mathematical purposes is the radical form \sqrt{69} .

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