Simplify square root of 216.
Understand the Problem
The question is asking us to simplify the square root of 216 to its simplest form. This involves breaking down the number into its prime factors and finding the largest perfect square that can be extracted from it.
Answer
6 \sqrt{6}
Answer for screen readers
The final answer is $6 \sqrt{6}$
Steps to Solve
- Prime factorization of 216
To break down 216 into its prime factors, we start by dividing by the smallest prime number (2) and continue:
$$ 216 \div 2 = 108 $$ $$ 108 \div 2 = 54 $$ $$ 54 \div 2 = 27 $$ $$ 27 \div 3 = 9 $$ $$ 9 \div 3 = 3 $$ $$ 3 \div 3 = 1 $$
So, the prime factorization of 216 is:
$$ 216 = 2^3 \times 3^3 $$
- Extract the largest perfect square
Identify pairs of primes to take outside the square root. For each pair of primes, one prime can be taken outside the square root:
$$ \sqrt{216} = \sqrt{2^3 \times 3^3} = \sqrt{(2^2 \times 3^2) \times (2 \times 3)} $$
Since $2^2$ and $3^2$ are perfect squares:
$$ = \sqrt{2^2 \times 3^2 \times 6} $$
- Simplify the square root
Take the square root of the perfect squares and leave the remaining factor inside the square root:
$$ = 2 \times 3 \times \sqrt{6} $$ $$ = 6\sqrt{6} $$
The final answer is $6 \sqrt{6}$
More Information
Breaking down numbers into prime factors is a valuable technique for simplifying square roots or finding the greatest common divisor.
Tips
A common mistake is not fully factoring the number or incorrectly identifying perfect squares. Double-check each factorization step to ensure accuracy.
AI-generated content may contain errors. Please verify critical information
