To simplify \(\sqrt{75}\), we need to follow a series of steps systematically.
1. Prime Factorization:
Begin by finding the prime factorization of 75.
\(75 = 3 \times 25\)
Further, 25 can be broken down into:
\(25 = 5 \times 5\)
Putting it all together:
\(75 = 3 \times 5^2\)
2. Expressing Under the Square Root:
Now we need to express \(\sqrt{75}\) using these prime factors.
\(\sqrt{75} = \sqrt{3 \times 5^2}\)
3. Breaking Down the Square Root:
The property of square roots we use here is \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). We apply this to our radicals:
\(\sqrt{75} = \sqrt{3 \times 5^2} = \sqrt{3} \times \sqrt{5^2}\)
4. Simplifying Further:
Recognize that \(\sqrt{5^2}\) simplifies to 5, because the square root of a square number returns the base:
\(\sqrt{75} = \sqrt{3} \times 5 = 5\sqrt{3}\)
Thus, the simplified form of \(\sqrt{75}\) is \(5\sqrt{3}\).
We can also verify this with the numerical value.
5. Verification:
- The actual value of \(\sqrt{75}\) is approximately 8.660254037844386.
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The simplified expression \(5\sqrt{3}\) should give us the same numerical value:
\(5 \times \sqrt{3} \approx 5 \times 1.7320508075688772 \approx 8.660254037844386\)
Therefore, [tex]\(\sqrt{75}\)[/tex] simplified is indeed [tex]\(5\sqrt{3}\)[/tex], and our step-by-step solution verifies this result.
