ماشین‌حساب ریشهٔ دوم و ساده‌سازی رادیکال
ریشهٔ دوم ۶۵ چند است؟ √65 ≈ 8.062258، و ساده نمی‌شود چون ۶۵ عامل مربعی ندارد. این ماشین‌حساب هر ریشه‌ای را پیدا می‌کند و آن را به ساده‌ترین شکل رادیکالی می‌نویسد، مثل √180 = 6√5.
نتیجه
مقادیر را وارد کنید و برای دیدن نتیجه «محاسبه» را بزنید.
فرمول
Take square factors out of the root: √72 = √(36 × 2) = 6√2.
For an nth root, take out factors that are perfect nth powers.
نحوه استفاده از این ماشین‌حساب
- Enter the number and choose which root you want.
- Press Calculate to get the simplified exact root and its decimal value.
مثال‌های حل‌شده
The square root of 72
72 = 36 × 2, and 36 = 6² is a perfect square.
√72 = √(6² × 2) = 6√2 ≈ 8.485281.
The square root of 65
65 = 5 × 13. Neither factor is a perfect square, so √65 is already in simplest radical form.
√65 lies between √64 = 8 and √81 = 9, very close to 8: √65 ≈ 8.062258.
Simplify a cube root: ∛54
Choose the cube root. 54 = 27 × 2, and 27 = 3³ is a perfect cube.
∛54 = ∛27 × ∛2 = 3∛2 ≈ 3.779763.
اشتباه‌های رایج
- Not using the largest square factor: √180 = 2√45 is true but unfinished — 45 still contains 9. The simplest form is 6√5.
- Splitting a sum: √(9 + 16) is 5, not √9 + √16 = 7.
- Treating the decimal as exact: √65 ≈ 8.062258 is rounded; √65 is the exact value.
پرسش‌های متداول
How do you simplify √180?
180 = 36 × 5, so √180 = 6√5 ≈ 13.416408 in simplest radical form.
What is the square root of 2116?
46, because 46 × 46 = 2116. It is a perfect square, so the root is a whole number.
What is the square root of x⁶?
√(x⁶) = |x³|, because (x³)² = x⁶. For x ≥ 0 that is simply x³. This calculator works with numbers; the algebra calculator handles expressions with letters.
Why is the answer written as 6√2 instead of a decimal?
√2 never ends or repeats, so 6√2 is the exact answer. The decimal shown with it is rounded.
Can I take the root of a negative number?
Odd roots, yes: ∛(−27) = −3. Even roots (square, 4th, …) of a negative number are not real numbers.