square root of 40 simplified radical form
Understand the Problem
The question is asking for the simplified form of the square root of 40. To solve this, we will factor 40 into its prime factors and simplify the radical accordingly.
Answer
The simplified form of the square root of 40 is \( 2\sqrt{10} \).
Answer for screen readers
The simplified form of the square root of 40 is ( 2\sqrt{10} ).
Steps to Solve
- Factor 40 into Prime Factors
To simplify the square root of 40, start by finding its prime factors.
The prime factorization of 40 is: $$ 40 = 2^3 \times 5 $$
- Apply the Product Property of Square Roots
Next, we can use the property of square roots that states ( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} ).
We can separate the square root of 40 into its prime factors: $$ \sqrt{40} = \sqrt{2^3 \times 5} = \sqrt{(2^2 \times 2) \times 5} = \sqrt{2^2} \times \sqrt{2} \times \sqrt{5} $$
- Simplify the Square Roots
Now, simplify ( \sqrt{2^2} ): $$ \sqrt{2^2} = 2 $$
So now we have: $$ \sqrt{40} = 2 \times \sqrt{2} \times \sqrt{5} = 2 \sqrt{10} $$
- Write the Final Answer
Thus, the simplified form of the square root of 40 is: $$ \sqrt{40} = 2\sqrt{10} $$
The simplified form of the square root of 40 is ( 2\sqrt{10} ).
More Information
The square root of 40 simplifies to ( 2\sqrt{10} ) because it includes the perfect square ( 2^2 ), allowing us to take it out of the square root. This technique is useful for simplifying square roots with prime factorization.
Tips
- Forgetting to factor completely can lead to a non-simplified final answer.
- Incorrectly applying the product property of square roots can result in errors when simplifying.
