Square root of 432 simplified.
Understand the Problem
The question is asking for the simplified form of the square root of 432. To simplify, we need to find the prime factorization of 432 and then simplify the square root accordingly.
Answer
$12\sqrt{3}$
Answer for screen readers
The simplified form of the square root of 432 is $12\sqrt{3}$.
Steps to Solve
- Find the prime factorization of 432
Start by dividing 432 by the smallest prime numbers until you reach 1.
- Divide by 2: $$ 432 \div 2 = 216$$
- Divide by 2 again: $$ 216 \div 2 = 108$$
- Divide by 2 again: $$ 108 \div 2 = 54$$
- Divide by 2 again: $$ 54 \div 2 = 27$$
- Divide by 3 (the next smallest prime): $$ 27 \div 3 = 9$$
- Divide by 3 again: $$ 9 \div 3 = 3$$
- Divide by 3 again: $$ 3 \div 3 = 1$$
So, the prime factorization of 432 is: $$ 432 = 2^4 \times 3^3 $$
- Apply the square root to the prime factorization
We can now apply the square root to each prime factor.
Using the property that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we get: $$ \sqrt{432} = \sqrt{2^4} \times \sqrt{3^3} $$
- Simplify each square root
Now calculate the square roots:
-
For $2^4$: $$ \sqrt{2^4} = 2^2 = 4 $$
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For $3^3$: $$ \sqrt{3^3} = \sqrt{3^2 \times 3} = 3 \times \sqrt{3} = 3 \sqrt{3} $$
- Combine the results
Now, combine the simplified parts together: $$ \sqrt{432} = 4 \times 3 \sqrt{3} = 12 \sqrt{3} $$
The simplified form of the square root of 432 is $12\sqrt{3}$.
More Information
The number 432 is interesting because it is a highly composite number, meaning it has more divisors than any smaller number. The simplification method shows how prime factorization helps in finding square roots.
Tips
- Forgetting to include all prime factors while factorizing the number.
- Not simplifying the square root correctly by missing out on perfect squares.
- Confusing the square root of a product with the product of square roots.
