What is the square root of 240 simplified?
Understand the Problem
The question is asking for the square root of 240 and wants it simplified. This involves factoring 240 into its prime factors and simplifying the square root accordingly.
Answer
The simplified square root of 240 is \( 4\sqrt{15} \).
Answer for screen readers
The simplified square root of 240 is ( 4\sqrt{15} ).
Steps to Solve
- Factor the Number First, we will factor 240 into its prime factors.
To factor 240, we can divide by the smallest prime numbers:
$$ 240 \div 2 = 120 \ 120 \div 2 = 60 \ 60 \div 2 = 30 \ 30 \div 2 = 15 \ 15 \div 3 = 5 \ 5 \div 5 = 1 $$
So, the prime factorization of 240 is
$$ 240 = 2^4 \times 3^1 \times 5^1 $$
- Set Up the Square Root Next, we express the square root of 240 using its prime factorization:
$$ \sqrt{240} = \sqrt{2^4 \times 3^1 \times 5^1} $$
- Simplify the Square Root Now we simplify the square root by using the property that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, and we know that $\sqrt{a^2} = a$:
$$ \sqrt{240} = \sqrt{(2^2)^2} \times \sqrt{3} \times \sqrt{5} $$ This simplifies to:
$$ \sqrt{240} = 2^2 \times \sqrt{15} = 4\sqrt{15} $$
The simplified square root of 240 is ( 4\sqrt{15} ).
More Information
The square root of 240 simplifies to ( 4\sqrt{15} ), where ( 15 ) cannot be simplified further in this context since it is not a perfect square. This method of factoring is useful for simplifying square roots of composite numbers.
Tips
- Not factoring properly: Make sure to fully factor the number into its prime components, as missing factors will lead to incorrect simplification.
- Forgetting square root properties: Remember to separate the square root into components properly; using properties like $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ is crucial.
