Last updated on May 26th, 2025
If a number is multiplied by itself, the result is a square. The inverse operation is finding the square root. Square roots have applications in fields like engineering, finance, and physics. Here, we will discuss the square root of 867.
The square root is the inverse operation of squaring a number. 867 is not a perfect square. The square root of 867 can be expressed in both radical and exponential forms. In radical form, it is expressed as √867, whereas in exponential form, it is (867)^(1/2). The approximate value of √867 is 29.445, which is an irrational number because it cannot be expressed as a fraction of two integers.
For perfect square numbers, the prime factorization method can be used. However, for non-perfect square numbers like 867, the long division and approximation methods are more suitable. Let us explore these methods: Prime factorization method Long division method Approximation method
The prime factorization of a number involves expressing it as a product of prime numbers. Let's examine the prime factorization of 867:
Step 1: Finding the prime factors of 867
Breaking it down, 867 = 3 × 17 × 17.
Step 2: Since 867 is not a perfect square, its digits cannot be grouped into pairs of equal factors. Thus, calculating √867 using prime factorization is not straightforward, but it helps in verifying results obtained by other methods.
The long division method is useful for non-perfect square numbers. Here's how to find the square root of 867 step by step:
Step 1: Begin by grouping the digits from right to left. For 867, we have 67 and 8.
Step 2: Find a number n whose square is less than or equal to 8. The number is 2 because 2 × 2 = 4. The quotient is 2, and the remainder is 8 - 4 = 4.
Step 3: Bring down the next group, 67, to make it 467. Add the previous divisor (2) to itself to get 4.
Step 4: Find a digit x such that 4x × x is less than or equal to 467. Let x be 9; then 49 × 9 = 441.
Step 5: Subtract 441 from 467; the remainder is 26.
Step 6: Add a decimal point, bring down 00 to make it 2600, and repeat the process.
Step 7: Continue this process to achieve the desired accuracy.
The quotient now is approximately 29.445.
The approximation method is a simpler way to estimate square roots. Here's how to approximate the square root of 867:
Step 1: Identify the perfect squares closest to 867. The perfect squares are 841 (29²) and 900 (30²), so √867 is between 29 and 30.
Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (867 - 841) / (900 - 841) = 26 / 59 ≈ 0.441
Step 3: Add this to the smaller root: 29 + 0.441 = 29.441
So the approximate square root of 867 is 29.441.
Students often make errors when calculating square roots, such as ignoring negative square roots or skipping steps in the long division method. Here are some common mistakes and how to avoid them:
Can you help Max find the area of a square box if its side length is given as √867?
The area of the square is 867 square units.
The area of the square = side².
The side length is given as √867.
Area = (√867)² = 867.
Therefore, the area of the square box is 867 square units.
A square-shaped building measuring 867 square feet is built; if each of the sides is √867, what will be the square feet of half of the building?
433.5 square feet
To find half of the building's area, divide the given area by 2. 867 / 2 = 433.5
So half of the building measures 433.5 square feet.
Calculate √867 × 5.
147.225
First, find the square root of 867, which is approximately 29.445.
Then multiply 29.445 by 5. 29.445 × 5 = 147.225
What will be the square root of (867 + 33)?
The square root is 30.
To find the square root, first calculate the sum (867 + 33) = 900, then find the square root of 900. √900 = 30.
Therefore, the square root of (867 + 33) is ±30.
Find the perimeter of the rectangle if its length ‘l’ is √867 units and the width ‘w’ is 40 units.
We find the perimeter of the rectangle as 138.89 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√867 + 40) Perimeter = 2 × (29.445 + 40) = 2 × 69.445 = 138.89 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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