Last updated on May 26th, 2025
The square root of 289 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 289. The number 289 has a unique non-negative square root, called the principal square root.
The square root of 289 is ±17, where 17 is the positive solution of the equation x2 = 289. Finding the square root is just the inverse of squaring a number and hence, squaring 17 will result in 289. The square root of 289 is written as √289 in radical form, where the ‘√’ sign is called the “radical” sign. In exponential form, it is written as (289)1/2
We can find the square root of 289 through various methods. They are:
The prime factorization of 289 can be found by dividing the number by prime numbers and continuing to divide the quotients until they can’t be separated anymore, i.e., we first prime factorize 289 and then make pairs of two to get the square root.
So, Prime factorization of 289 = 17 ×17
Square root of 289= √[17 × 17] = 17
This method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.
Follow the steps to calculate the square root of 289:
Step 1: Write the number 289 and draw a bar above the pair of digits from right to left.
Step 2: Now, find the greatest number whose square is less than or equal to 2. Here, it is1 because 12=1 < 2
Step 3: now divide 289 by 1 (the number we got from Step 2) such that we get 1 as a quotient, and we get a remainder. Double the divisor 1, we get 2, and then the largest possible number A1=7 is chosen such that when 7 is written beside the new divisor 2, a 2-digit number is formed →27, and multiplying 7 with 27 gives 189, which
when subtracted from 189, gives 0
Repeat this process until you reach the remainder of 0.
Step 4: The quotient obtained is the square root of 289. In this case, it is 17.
We know that the sum of the first n odd numbers is n2. We will use this fact to find square roots through the repeated subtraction method. Furthermore, we just have to subtract consecutive odd numbers from the given number, starting from 1. The square root of the given number will be the count of the number of steps required to obtain 0. Here are the steps:
Step 1: take the number 289 and then subtract the first odd number from it. Here, in this case, it is 289-1=288
Step 2: we have to subtract the next odd number from the obtained number until it comes zero as a result. Now take the obtained number (from Step 1), i.e., 288, and again subtract the next odd number after 1, which is 3, → 288-3=285. Like this, we have to proceed further.
Step 3: now we have to count the number of subtraction steps it takes to yield 0 finally. Here, in this case, it takes 17 steps.
So, the square root is equal to the count, i.e., the square root of 289 is ±17.
When we find the square root of 289, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
Find √(289⤬256) ?
√(289⤬256)
= 17 ⤬16
= 272
Answer : 272
firstly, we found the values of the square roots of 289 and 256, then multiplied the values.
What is √289 multiplied by 17 ?
√289 ⤬ 17
= 17⤬17
= 289
Answer: 289
finding the value of √289 and multiplying by 17.
Find the radius of a circle whose area is 289π cm².
Given, the area of the circle = 289π cm2
Now, area = πr2 (r is the radius of the circle)
So, πr2 = 289π cm2
We get, r2 = 289 cm2
r = √289 cm
Putting the value of √289 in the above equation,
We get, r = ±17 cm
Here we will consider the positive value of 17.
Therefore, the radius of the circle is 17 cm.
Answer: 17 cm
We know that, area of a circle = πr2 (r is the radius of the circle).According to this equation, we are getting the value of “r” as 17 cm by finding the value of the square root of 289
Find the length of a side of a square whose area is 289 cm²
Given, the area = 289 cm2
We know that, (side of a square)2 = area of square
Or, (side of a square)2 = 289
Or, (side of a square)= √289
Or, the side of a square = ± 17.
But, the length of a square is a positive quantity only, so, the length of the side is 17 cm.
Answer: 17 cm
We know that, (side of a square)2 = area of square. Here, we are given with the area of the square, so, we can easily find out its square root because its Square root is the measure of the side of the square
Find √289 / √49
√289/√49
= 17/7
= 2.429
Answer : 2.429
we firstly found out the values of √289 and √49, then divided .
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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