CalcOpenly
ಕನ್ನಡ

ಘಾತ ಮತ್ತು ಮೂಲ ಕ್ಯಾಲ್ಕುಲೇಟರ್

Raise a number to any exponent, including fractional and negative ones, or take its square, cube or nth root, with exact simplified radicals such as 5√2.

ನವೀಕರಿಸಲಾಗಿದೆ ಪರಿಶೀಲಿಸಿದ ಉದಾಹರಣೆಗಳು: 9

Integers, decimals, fractions (27/8) or expressions
A fraction like 2/3 means the cube root, squared
ಪ್ರಯತ್ನಿಸಿ
ಫಲಿತಾಂಶ
ಫಲಿತಾಂಶ: 6.349604207873
ಗರಿಷ್ಠ ದಶಮಾಂಶ ಸ್ಥಾನಗಳು: 12; ಹತ್ತಿರದ ಮೌಲ್ಯಕ್ಕೆ; ಸಮದೂರದಲ್ಲಿದ್ದರೆ ಸೊನ್ನೆಯಿಂದ ದೂರಕ್ಕೆ
Exact form
4∛4

16 to the power 2/3 is 4∛4 ≈ 6.34960420787.

y = t^(2/3)

2468-20-1001020ty16 → 6.3496
ಲೆಕ್ಕಾಚಾರದ ವಿಧಾನ S

  1. Fractional exponent as a root

    1623=162316^{\frac{2}{3}} = \sqrt[3]{16^{2}}

    The denominator 3 is the root; the numerator 2 is the power.

  2. Raise to the power first

    162=25616^{2} = 256

  3. Take out perfect cubes

    2563=43⋅43=443\sqrt[3]{256} = \sqrt[3]{4^{3} \cdot 4} = 4\sqrt[3]{4}

  4. Decimal value

    443≈6.3496042078727978994\sqrt[3]{4} \approx 6.349604207872797899

ಘಾತ ಮತ್ತು ಮೂಲ ಕ್ಯಾಲ್ಕುಲೇಟರ್ ಬಗ್ಗೆ

An exponent says how many times to multiply a base by itself: 2⁵ = 32. The rules extend it beyond whole numbers. A negative exponent is a reciprocal (2⁻³ = 1/8), and a fractional exponent m/k is a root, x^(m/k) = (ᵏ√x)^m, so 8^(2/3) = (∛8)² = 4. Roots are simplified by pulling out perfect powers: √50 = √(25 × 2) = 5√2.

Algebra homework, simplifying radicals and formulas with fractional powers are the usual uses. The default, 16^(2/3), is ∛256 = 4∛4 ≈ 6.3496, and the cube root of 54 is 3∛2 because 54 = 27 × 2.

The answer is exact whenever an exact form exists, as a fraction or a simplified radical, and the decimal value is shown to 12 places. Odd roots of negative numbers are real, (−8)^(1/3) = −2; even roots of negative numbers have no real value. 0⁰ is taken as 1.

ಪರಿಹರಿಸಿದ ಉದಾಹರಣೆಗಳು

8^(2/3)

Calculate
Power xʸ
Base x
8
Exponent y
2/3
ಫಲಿತಾಂಶ
4
Exact form
4

ಪರಿಶೀಲನೆಯ ಮೂಲ: ∛8 = 2, 2² = 4 (hand calculation; Python 8 ** (2/3) = 3.9999999999999996)

Cube root of 54

Calculate
Root ⁿ√x
Number x
54
Root n
3
ಫಲಿತಾಂಶ
3.779763149685
Exact form
3∛2

ಪರಿಶೀಲನೆಯ ಮೂಲ: 54 = 27 × 2; Python decimal (60 digits) Decimal(54) ** (Decimal(1)/3) = 3.7797631496846193…

2^−3

Calculate
Power xʸ
Base x
2
Exponent y
-3
ಫಲಿತಾಂಶ
0.125
Exact form
1/8

ಪರಿಶೀಲನೆಯ ಮೂಲ: Python Fraction(2) ** -3 = 1/8

Negative base, odd root: (−8)^(1/3)

Calculate
Power xʸ
Base x
-8
Exponent y
1/3
ಫಲಿತಾಂಶ
-2
Exact form
−2

ಪರಿಶೀಲನೆಯ ಮೂಲ: (−2)³ = −8 (real cube root)

ಪ್ರಶ್ನೆಗಳು

What does a fractional exponent mean?

The denominator is a root and the numerator is a power: x^(m/k) = (ᵏ√x)^m. So 8^(2/3) takes the cube root of 8, which is 2, and squares it to give 4, while 16^(1/2) = √16 = 4. A decimal exponent works the same way once written as a fraction: 2^0.5 = 2^(1/2) = √2 ≈ 1.41421.

What does a negative exponent mean?

A negative exponent means the reciprocal of the positive power: x⁻ⁿ = 1/xⁿ. So 2⁻³ = 1/2³ = 1/8 = 0.125, and 10⁻² = 0.01. The rule follows from dividing powers, 2³ ÷ 2⁶ = 2³⁻⁶ = 2⁻³. A negative exponent never makes the result negative, and 0 raised to a negative power is undefined because it means dividing by zero.

How do you simplify a square root?

Split the number into the largest perfect square times what is left, then take the square root of the square. 50 = 25 × 2, so √50 = 5√2 ≈ 7.0711. Higher roots work the same way with perfect cubes, fourth powers and so on: 54 = 27 × 2, so ∛54 = 3∛2. The radical is fully simplified when nothing left under the sign has such a factor, as with √2 or ∛4.

Why is 0 to the power 0 equal to 1?

By convention x⁰ = 1 for every x, including 0, because that keeps formulas such as the binomial theorem and the power series eˣ = Σ xⁿ/n! correct at x = 0; Knuth's Concrete Mathematics (§5.1) argues for this definition. As a limit, though, 0⁰ is an indeterminate form: xʸ can approach different values as x and y both approach 0.

Can you take the square root of a negative number?

Not as a real number, because no real number squared gives −4. Odd roots are different: a negative number cubed stays negative, so ∛(−8) = −2 and the fifth root of −32 is −2. Even roots of negative numbers exist only as complex numbers, √(−4) = 2i, which the complex number calculator handles.

“ಘಾತ ಮತ್ತು ಮೂಲ ಕ್ಯಾಲ್ಕುಲೇಟರ್” ಎಷ್ಟು ನಿಖರವಾಗಿದೆ?

ನಿಖರತೆ ನಿಮ್ಮ ಇನ್‌ಪುಟ್‌ಗಳು ಮತ್ತು ವಿಧಾನದ ಊಹೆಗಳನ್ನು ಅವಲಂಬಿಸಿದೆ. ದಶಮಾಂಶ ಗಣನೆ 50 ಸಾರ್ಥಕ ಅಂಕೆಗಳನ್ನು ಬಳಸುತ್ತದೆ. ಆದರೆ ಅಂದಾಜುಗಳು, ಸಂಖ್ಯಾತ್ಮಕ ವಿಧಾನಗಳು ಮತ್ತು ಮೂಲ ದತ್ತಾಂಶ ಕಡಿಮೆ ನಿಖರವಾಗಿರಬಹುದು; ಪ್ರದರ್ಶಿತ ಮೌಲ್ಯಗಳನ್ನು ರೌಂಡ್ ಮಾಡುವುದರಿಂದ ಈ ಮಿತಿಗಳು ನಿವಾರಣೆಯಾಗುವುದಿಲ್ಲ. ಸ್ವತಂತ್ರ ಮೂಲಗಳ ಪರಿಹಾರಗಳೊಂದಿಗೆ ಪರಿಶೀಲಿಸಿದ ಉದಾಹರಣೆಗಳು: 9. ಉದಾಹರಣೆಗೆ, “8^(2/3)” ಅನ್ನು ∛8 = 2, 2² = 4 (hand calculation; Python 8 ** (2/3) = 3.9999999999999996) ಜೊತೆಗೆ ಪರಿಶೀಲಿಸಲಾಗುತ್ತದೆ.

ಈ ವಿಧಾನದ ಮೂಲ ಯಾವುದು?

Khan Academy — Rational exponents and radicals; Wolfram MathWorld — Radical.

ಈ ಕ್ಯಾಲ್ಕುಲೇಟರ್ ಬಗ್ಗೆ

xm/k=(xk)m=xmkakrk=ark\begin{gathered} x^{m/k} = \left(\sqrt[k]{x}\right)^{m} = \sqrt[k]{x^{m}} \\[10pt] \sqrt[k]{a^{k} r} = a\sqrt[k]{r} \end{gathered}

ಮೂಲಗಳು

  1. Khan Academy — Rational exponents and radicals
  2. Wolfram MathWorld — Radical

ಆಧಾರಗಳೊಂದಿಗೆ ಪರಿಶೀಲಿಸಲಾಗಿದೆ

ಈ ಕ್ಯಾಲ್ಕುಲೇಟರ್‌ನಲ್ಲಿ ಸ್ವತಂತ್ರ ಆಧಾರಗಳಿಂದ ಉತ್ತರಗಳನ್ನು ಪಡೆದ 9 ಬಿಡಿಸಿದ ಉದಾಹರಣೆಗಳಿವೆ. ಇವು ಪರೀಕ್ಷಾ ಸಮೂಹದಲ್ಲಿ ನಡೆಯುತ್ತವೆ; ನೀವು ಇಲ್ಲಿಯೂ ನಡೆಸಬಹುದು.

ಸಂಬಂಧಿತ ಕ್ಯಾಲ್ಕುಲೇಟರ್‌ಗಳು