Square root of 75 simplified radical form
Understand the Problem
The question is asking for the square root of 75 to be expressed in its simplified radical form, which involves breaking down the number into its prime factors and simplifying accordingly.
Answer
$5\sqrt{3}$
Answer for screen readers
The simplified radical form of the square root of 75 is $5\sqrt{3}$.
Steps to Solve
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Factor 75 into its prime components First, identify the prime factors of 75. We can do this by dividing by the smallest prime numbers: $$ 75 = 3 \times 25 $$ Next, factor 25 further: $$ 25 = 5 \times 5 $$ So, the complete factorization of 75 is: $$ 75 = 3 \times 5^2 $$
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Apply the square root to the factors Now we can express the square root of 75 using the prime factors: $$ \sqrt{75} = \sqrt{3 \times 5^2} $$
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Simplify using properties of square roots We can separate the square root and simplify: $$ \sqrt{75} = \sqrt{3} \times \sqrt{5^2} $$ Since the square root of $5^2$ is 5, we then have: $$ \sqrt{75} = \sqrt{3} \times 5 $$
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Write the final simplified form Rearranging gives us: $$ \sqrt{75} = 5\sqrt{3} $$
The simplified radical form of the square root of 75 is $5\sqrt{3}$.
More Information
The square root of 75 can be simplified by breaking it down to its prime factors, which allows us to express it in a more manageable form. Understanding how to simplify square roots is a useful skill in algebra and number theory.
Tips
- Forgetting to fully factor the number into prime factors. It's important to break down each composite number completely.
- Failing to simplify the square root correctly, especially in identifying perfect squares among the factors.
