Square Root of 178
The square root of 178 is expressed as √178 in the radical form and as (178)½ or (178)0.5 in the exponent form. The square root of 178 rounded up to 10 decimal places is 13.3416640641. It is the positive solution of the equation x2 = 178.
- Square Root of 178: 13.341664064126334
- Square Root of 178 in exponential form: (178)½ or (178)0.5
- Square Root of 178 in radical form: √178
1. | What is the Square Root of 178? |
2. | How to find the Square Root of 178? |
3. | Is the Square Root of 178 Irrational? |
4. | FAQs |
What is the Square Root of 178?
The square root of 178, (or root 178), is the number which when multiplied by itself gives the product as 178. Therefore, the square root of 178 = √178 = 13.341664064126334.
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How to Find Square Root of 178?
Value of √178 by Long Division Method
Explanation:
- Forming pairs: 01 and 78
- Find a number Y (1) such that whose square is <= 1. Now divide 01 by 1 with quotient as 1.
- Bring down the next pair 78, to the right of the remainder 0. The new dividend is now 78.
- Add the last digit of the quotient (1) to the divisor (1) i.e. 1 + 1 = 2. To the right of 2, find a digit Z (which is 3) such that 2Z × Z <= 78. After finding Z, together 2 and Z (3) form a new divisor 23 for the new dividend 78.
- Divide 78 by 23 with the quotient as 3, giving the remainder = 78 - 23 × 3 = 78 - 69 = 9.
- Now, let's find the decimal places after the quotient 13.
- Bring down 00 to the right of this remainder 9. The new dividend is now 900.
- Add the last digit of quotient to divisor i.e. 3 + 23 = 26. To the right of 26, find a digit Z (which is 3) such that 26Z × Z <= 900. Together they form a new divisor (263) for the new dividend (900).
- Divide 900 by 263 with the quotient as 3, giving the remainder = 900 - 263 × 3 = 900 - 789 = 111.
- Bring down 00 again. Repeat above steps for finding more decimal places for the square root of 178.
Therefore, the square root of 178 by long division method is 13.3 approximately.
Is Square Root of 178 Irrational?
The actual value of √178 is undetermined. The value of √178 up to 25 decimal places is 13.34166406412633371248944. Hence, the square root of 178 is an irrational number.
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- Square Root of 49 - √49 = 7
- Square Root of 676 - √676 = 26
- Square Root of 36 - √36 = 6
- Square Root of 25 - √25 = 5
Square Root of 178 Solved Examples
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Example 1: Solve the equation x2 − 178 = 0
Solution:
x2 - 178 = 0 i.e. x2 = 178
x = ±√178
Since the value of the square root of 178 is 13.342,
⇒ x = +√178 or -√178 = 13.342 or -13.342. -
Example 2: If the area of an equilateral triangle is 178√3 in2. Find the length of one of the sides of the triangle.
Solution:
Let 'a' be the length of one of the sides of the equilateral triangle.
⇒ Area of the equilateral triangle = (√3/4)a2 = 178√3 in2
⇒ a = ±√712 in
Since length can't be negative,
⇒ a = √712 = 2 √178
We know that the square root of 178 is 13.342.
⇒ a = 26.683 in -
Example 3: If the surface area of a sphere is 712π in2. Find the radius of the sphere.
Solution:
Let 'r' be the radius of the sphere.
⇒ Area of the sphere = 4πr2 = 712π in2
⇒ r = ±√178 in
Since radius can't be negative,
⇒ r = √178
The square root of 178 is 13.342.
⇒ r = 13.342 in
FAQs on the Square Root of 178
What is the Value of the Square Root of 178?
The square root of 178 is 13.34166.
Why is the Square Root of 178 an Irrational Number?
Upon prime factorizing 178 i.e. 21 × 891, 2 is in odd power. Therefore, the square root of 178 is irrational.
What is the Square Root of -178?
The square root of -178 is an imaginary number. It can be written as √-178 = √-1 × √178 = i √178 = 13.341i
where i = √-1 and it is called the imaginary unit.
What is the Square Root of 178 in Simplest Radical Form?
We need to express 178 as the product of its prime factors i.e. 178 = 2 × 89. Therefore, as visible, the radical form of the square root of 178 cannot be simplified further. Therefore, the simplest radical form of the square root of 178 can be written as √178
Evaluate 13 plus 8 square root 178
The given expression is 13 + 8 √178. We know that the square root of 178 is 13.342. Therefore, 13 + 8 √178 = 13 + 8 × 13.342 = 13 + 106.733 = 119.733
If the Square Root of 178 is 13.342. Find the Value of the Square Root of 1.78.
Let us represent √1.78 in p/q form i.e. √(178/100) = 1.78/10 = 1.334. Hence, the value of √1.78 = 1.334
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