sqrt 180 simplified

Understand the Problem

The question is asking for the simplified form of the square root of 180. This involves finding the prime factorization of 180 and simplifying the square root accordingly.

Answer

The simplified form of the square root of 180 is \( 6\sqrt{5} \).

Steps to Solve

Find the prime factorization of 180

Start by breaking down the number 180 into its prime factors.

[ 180 = 2 \times 90 ]

Next, continue factoring 90 into primes:

[ 90 = 2 \times 45 ]

Then, factor 45:

[ 45 = 3 \times 15 ]

And factor 15:

[ 15 = 3 \times 5 ]

So, the complete prime factorization of 180 is:

[ 180 = 2^2 \times 3^2 \times 5 ]

Apply the square root to the prime factorization

Now, we take the square root of 180 using its prime factorization:

[ \sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} ]

Separate the perfect squares from the remaining factors

We can split the square root into the product of square roots of the perfect squares:

[ \sqrt{180} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{5} ]

Simplify the square root of perfect squares

Now, simplify the square roots of the perfect squares:

[ \sqrt{2^2} = 2 \quad \text{and} \quad \sqrt{3^2} = 3 ]

So, we combine these results:

[ \sqrt{180} = 2 \times 3 \times \sqrt{5} ]

Final simplification

Now, multiply the rational parts:

[ \sqrt{180} = 6 \sqrt{5} ]

The simplified form of the square root of 180 is ( 6\sqrt{5} ).

More Information

The square root of a number can often be simplified by finding its prime factors. This process shows how factors can combine to simplify expressions. This approach is especially useful for square roots of non-perfect squares.

Tips

Forgetting to factor completely before taking the square root. Make sure to break down the number fully into primes.
Confusing square root simplification rules. Remember only pairs of prime factors under the square root can be simplified.

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