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Last updated on April 28th, 2025
If a number is multiplied by itself, the result is a square. The inverse operation is finding the square root. The square root has applications in various fields such as engineering, physics, and finance. Here, we will discuss the square root of 167.
The square root is the inverse operation of squaring a number. 167 is not a perfect square. The square root of 167 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √167, whereas in the exponential form, it is expressed as (167(1/2) .The approximate value of √167 is 12.9228, which is an irrational number because it cannot be expressed as a fraction of two integers.
For perfect square numbers, the prime factorization method is effective. However, for non-perfect squares like 167, the long division method and approximation method are more suitable. Let's explore these methods:
Prime factorization involves expressing a number as a product of prime numbers.
For 167, the prime factorization is straightforward since 167 is a prime number itself. Thus, it cannot be factored further.
Since 167 is not a perfect square, we cannot pair its prime factors to simplify the square root. Therefore, calculating √167 using prime factorization is not feasible.
The long division method is used for finding the square roots of non-perfect square numbers. Here’s how it works for 167:
Step 1: Group the number from right to left. In the case of 167, we group it as (1)(67).
Step 2: Find a number n whose square is less than or equal to 1. Here, n is 1 since 1^2 = 1. The quotient becomes 1, and the remainder is 0.
Step 3: Bring down the next group, 67, making the new dividend 67. Add the previous divisor (1) to itself to get 2, which is part of the new divisor.
Step 4: Consider 2n as the new divisor. We need to find n such that 2n × n ≤ 67. Trying n as 3 gives 23 × 3 = 69, which is too large. Trying n as 2 gives 22 × 2 = 44, which fits.
Step 5: Subtract 44 from 67 to get a remainder of 23.
Step 6: Since the new dividend is smaller than the divisor, add a decimal point and bring down two zeros to make it 2300.
Step 7: Find the new divisor, which becomes 249 (since the previous quotient was 12). Find n such that 249n × n ≤ 2300. Trying n as 9 gives 2499 × 9 = 2241.
Step 8: Subtract 2241 from 2300 to get a remainder of 59.
Step 9: Continue this process to get more decimal places as needed.
Thus, √167 ≈ 12.9228.
Approximation is a simpler method to estimate square roots. Follow these steps for √167:
Step 1: Identify the closest perfect squares. 144 and 169 are the nearest perfect squares to 167. √167 lies between √144 (12) and √169 (13).
Step 2: Use the formula to approximate: (Given number - smaller perfect square) / (Greater perfect square - smaller perfect square). For 167, (167 - 144) / (169 - 144) = 23 / 25 = 0.92. Step 3: Add the approximation to the smaller square root value: 12 + 0.92 = 12.92.
Therefore, √167 is approximately 12.92.
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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.
: He loves to play the quiz with kids through algebra to make kids love it.