Square root of 384 simplified

Understand the Problem

The question is asking for the square root of 384 to be simplified. This involves breaking down the number into its prime factors and expressing it in its simplest radical form.

Answer

The square root of 384 simplified is $ 8\sqrt{6} $.

Steps to Solve

Find the prime factorization of 384

To simplify the square root of 384, we first need to find its prime factorization.

Start dividing 384 by the smallest prime number, which is 2.
$384 \div 2 = 192$
$192 \div 2 = 96$
$96 \div 2 = 48$
$48 \div 2 = 24$
$24 \div 2 = 12$
$12 \div 2 = 6$
$6 \div 2 = 3$
$3$ is a prime number.

The complete factorization is: $$ 384 = 2^7 \times 3^1 $$

Rearranging prime factors for simplification

Next, we will rearrange the prime factors to group them into pairs, since we are finding a square root.

From $2^7$, we can take out $2^3$ from the square root because there are three pairs of 2s ($2^2 \times 2^2 \times 2^2$).
This leaves us with one 2 inside the radical.

Putting it together: $$ \sqrt{384} = \sqrt{2^6 \times 2^1 \times 3} = \sqrt{(2^3)^2 \times 2^1 \times 3} $$

Taking the square root

Now we can take the square root of the perfect squares.

The square root of $(2^3)^2$ is $2^3 = 8$.

Hence, we have: $$ \sqrt{384} = 8 \sqrt{6} $$

The simplified form of the square root of 384 is ( 8\sqrt{6} ).

More Information

The process of simplifying square roots by using prime factorization can be very helpful in many areas of math and science. Understanding how to manipulate radicals is a fundamental skill, especially in algebra.

Tips

Forgetting to pair the prime factors correctly when simplifying square roots can lead to incorrect answers. Always look for complete pairs.
Ignoring the remaining prime factors after extracting pairs may cause confusion about the final result.

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