Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in fields such as vehicle design, finance, etc. Here, we will discuss the square root of 170000.
The square root is the inverse of the square of a number. 170000 is not a perfect square. The square root of 170000 can be expressed in both radical and exponential forms. In radical form, it is expressed as √170000, whereas (170000)^(1/2) in exponential form. √170000 ≈ 412.3106, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers, where long-division and approximation methods are used. Let us now learn the following methods:
The product of prime factors is the prime factorization of a number. Now let us look at how 170000 is broken down into its prime factors.
Step 1: Finding the prime factors of 170000. Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 5 x 17 x 17: 2^3 x 5^3 x 17^2.
Step 2: Now we found out the prime factors of 170000. The second step is to make pairs of those prime factors. Since 170000 is not a perfect square, the digits of the number can’t be grouped into complete pairs. Therefore, calculating 170000 using prime factorization requires further steps, leading to an approximate square root.
The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.
Step 1: To begin with, we need to group the numbers from right to left. In the case of 170000, we can group it as 00, 00, and 17.
Step 2: Now we need to find n whose square is less than or equal to 17. We can say n is 4 because 4 x 4 = 16. Now the quotient is 4, and after subtracting 16 from 17, the remainder is 1.
Step 3: Bring down the next pair of digits (00) to get 100. Add 4 to the old divisor (4) to get 8, which will be our new divisor (80).
Step 4: Find the largest number x such that 80x x x ≤ 100. The largest such x is 1, since 801 x 1 = 80.
Step 5: Subtract 80 from 100 to get 20. Bring down the next pair of zeros (00) to get 2000.
Step 6: Continue the process until you achieve the desired decimal places. So the approximate square root of √170000 is 412.3106.
The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 170000 using the approximation method.
Step 1: Identify the closest perfect squares around 170000. The smallest perfect square less than 170000 is 169000 (410^2), and the largest perfect square greater than 170000 is 171396 (414^2).
Step 2: Use the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (170000 - 169000) / (171396 - 169000) = 1000 / (2396) ≈ 0.4173. Adding the approximate decimal to the closest smaller whole number, we get 410 + 0.4173 ≈ 412.3106.
Students may make mistakes while finding the square root, such as forgetting about the negative square root, skipping steps in the long division method, etc. Let us look at a few of these mistakes in detail.
Can you help Max find the area of a square plot if its side length is given as √170000?
The area of the square plot is 170000 square units.
The area of the square = side².
The side length is given as √170000.
Area of the square = side² = √170000 x √170000 = 170000.
Therefore, the area of the square plot is 170000 square units.
A square-shaped building measuring 170000 square feet is built; if each of the sides is √170000, what will be the square feet of half of the building?
85000 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 170000 by 2 = 85000.
So half of the building measures 85000 square feet.
Calculate √170000 x 5.
2061.553
The first step is to find the square root of 170000, which is approximately 412.3106.
The second step is to multiply 412.3106 by 5.
So, 412.3106 x 5 ≈ 2061.553.
What will be the square root of (160000 + 10000)?
The square root is approximately 412.3106.
To find the square root, we need to find the sum of (160000 + 10000).
160000 + 10000 = 170000. The square root of 170000 is approximately 412.3106.
Therefore, the square root of (160000 + 10000) is approximately ±412.3106.
Find the perimeter of a rectangle if its length 'l' is √170000 units and the width 'w' is 100 units.
The perimeter of the rectangle is approximately 1024.6212 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√170000 + 100) = 2 × (412.3106 + 100) = 2 × 512.3106 ≈ 1024.6212 units.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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