How to Find the Square Root of 58? 

How to Find the Square Root of 58? 

To find the square root of 58, you can use an approximation method. Start with an initial guess, like 7, calculate its square (49), and iteratively refine the guess until it’s close enough to 58. The square root of 58 is approximately 7.615 when rounded to three decimal places.

Certainly, here’s a table showing the square root of 58 to a few decimal places using long division:

ApproximationCalculationSquare of ApproximationDifference from 58
77 * 7 = 4949-9
7.57.5 * 7.5 = 56.2556.25-1.75
7.67.6 * 7.6 = 57.7657.76-0.24
7.6157.615 * 7.615 ≈ 58.0158.010.01

Using long division and approximations, we can see that the square root of 58 is approximately 7.615 to two decimal places.

Finding square roots of numbers is a fundamental mathematical skill with many applications in algebra, geometry, trigonometry and beyond. Square roots involve determining which number multiplied by itself equals the given number.

In this post, we’ll find the exact square root of 58 as well as estimate its square root. We’ll break down step-by-step how to calculate square roots and also discuss techniques for determining perfect squares and approximating roots for non-perfect squares.

Understanding Square Roots

Let’s start with a quick refresher on square roots. The square root of a number n is defined as the number that, when multiplied by itself, equals n. This relationship is denoted by the radical symbol:


For example:

√36 = 6 because 6 x 6 = 36 √49 = 7 because 7 x 7 = 49

Only numbers that are perfect squares have exact square root values that are integers. Non-perfect squares have irrational square roots that require approximation or using radicals.

Determining Perfect Squares

To find square roots, we first want to determine if the number is a perfect square. Some ways to identify perfect squares:

  • Look for numbers ending in 0, 1, 4, 5, 6, 9. These digit endings often indicate squares.
  • Break the number into factors and calculate the square root of each. 58 = (2 x 29), so √58 is between √4 = 2 and √81 = 9.
  • Refer to a perfect squares chart listing squares up to at least 100.

Using these methods, we determine 58 is not a perfect square. So √58 will be an irrational number that must be estimated.

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Estimating the Square Root of 58

Since 58 is between 49 and 64, two perfect squares, the square root of 58 is between their roots:

√49 = 7 √64 = 8

Therefore, √58 is approximately:

√58 ≈ 7.5

We can refine the estimate further by taking the average of 7 and 8:

(7 + 8) / 2 = 7.5

So a good estimate for √58 is 7.5.

Finding the Exact Value

To find the exact value for √58, we can use a calculator, which gives:

√58 = 7.6158

Or, we can leave it in exact radical form:



In this post, we walked through steps to determine the perfect square nearest to 58 and estimate √58 as 7.5. We also discussed strategies for identifying perfect squares and approximating non-perfect square roots. Finding square roots relies on recalling key concepts like perfect squares, factors, radicals, and simplification rules.

Regular practice builds skill in efficient root determination. Whether estimating roots or solving radicals, mastery of square roots provides a foundation for higher math and STEM applications. Keep pushing your square root skills forward!

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