What is the square root of 140 simplified?
Understand the Problem
The question is asking to simplify the square root of 140. To do this, we will look for the prime factors of 140 and determine if any perfect squares can be factored out.
Answer
$2\sqrt{35}$
Answer for screen readers
The simplified form of the square root of 140 is $2\sqrt{35}$.
Steps to Solve
- Find the prime factorization of 140
First, we find the prime factors of 140. Start by dividing by the smallest prime number, which is 2. $$ 140 \div 2 = 70 $$ Next, continue factoring 70. $$ 70 \div 2 = 35 $$ Now, we factor 35 by dividing by the next prime number, which is 5. $$ 35 \div 5 = 7 $$ Thus, the complete prime factorization of 140 is: $$ 140 = 2^2 \times 5 \times 7 $$
- Identify perfect squares
From the factorization, we can see that $2^2$ is a perfect square. We will take the square root of the perfect square and leave the rest under the square root. $$ \sqrt{140} = \sqrt{2^2 \times 5 \times 7} $$ We can simplify this to: $$ \sqrt{140} = 2 \times \sqrt{5 \times 7} $$
- Multiply the remaining square root
Now, calculate the remaining expression under the square root. $$ 5 \times 7 = 35 $$ So, we can simplify further: $$ \sqrt{140} = 2 \times \sqrt{35} $$
The simplified form of the square root of 140 is $2\sqrt{35}$.
More Information
The square root of a number can often be simplified by factoring it into prime components. In this case, recognizing and removing perfect squares simplifies the expression significantly.
Tips
- Forgetting to take out all the perfect squares. Always check for multiple perfect square factors.
- Assuming that the square root of a product can just be separated without simplifying fully. Make sure to simplify each factor correctly.
