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The nth Root and Fractional Indices

Learn what an nth root is and how to simplify fractional indices, like 12513=1253=5125^{\tfrac{1}{3}} = \sqrt[3]{125} = 5. Let’s get started! πŸš€

The nth Root and Fractional Indices - introduction visual

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Diagram explaining square, cube, and fourth roots. It shows the corresponding example calculations, with corresponding powers.Cube root of 8 equals 2, showing nth root rule: multiplying n roots of a number returns the original value.Cube root of 8 equals 2, showing βˆ›8Β·βˆ›8Β·βˆ›8 = 8 and fractional indices rule a^(1/n) = ⁿ√a.Demonstration of the cube root of 125, showing that 125 raised to the power of 1/3 equals 5, alongside an illustration of 5 cubed equalling 125.Fractional indices concept showing how to simplify 125^(2/3) = 25 using roots and powers, alongside the general formula a^(m/n) = (n√a)^m.Equation showing fractional indices, with 16 raised to the power of 3/4 equalling the cube of the fourth root of 16, simplified to 2, resulting in 8.

πŸ›ŽοΈ nα΅—Κ° Root

  • The nα΅—Κ° root means a number that is multiplied by itself nn times.
  • For example, the cube root of 2727 is 33 because 3Γ—3Γ—3=273 \times 3 \times 3 = 27.

πŸ›ŽοΈ Rules of the nα΅—Κ° Root

  • The algebraic rule is anΓ—anΓ—β‹―=a\sqrt[n]{a} \times \sqrt[n]{a} \times \dots = a when multiplied nn times.
  • For example, 83Γ—83Γ—83=8\sqrt[3]{8} \times \sqrt[3]{8} \times \sqrt[3]{8} = 8.

πŸ›ŽοΈ Roots as Fractional Indices

  • A fractional index is another way to write a root.
  • The rule is a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}

πŸ›ŽοΈ Example: 12513125^{\frac{1}{3}}

  • The fractional index 13\frac{1}{3} means take the cube root.
  • 12513=1253=5125^{\frac{1}{3}} = \sqrt[3]{125} = 5 because 53=1255^3 = 125.

πŸ›ŽοΈ Fractional Indices Rule

  • In amna^{\frac{m}{n}}, the denominator tells you the root.
  • The numerator tells you the power.

πŸ›ŽοΈ Example: 163416^{\frac{3}{4}}

  • Find the 4th root of 1616 to get 22.
  • Then raise it to the power 33 to get 88.

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Simplifying nth roots and fractional indices

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