The nth Root and Fractional Indices

Q: What is the nth root of a number?

The nth root of a number is the value that, multiplied by itself n times, gives that number. For example, the cube root of 8 is 2, because 2 cubed equals 8. The small number in the root symbol tells you which root to take.

Q: How do you find the nth root of a number?

To find the nth root, ask which number, raised to that power, gives the original number. For the cube root of 8, you look for the number that cubes to 8. That number is 2, because 2 times 2 times 2 equals 8.

Q: What does a power of one-third mean?

A power of one-third means the same as the cube root. Raising a number to the power of 1 over n is the same as taking its nth root. So 8 to the power of one-third equals the cube root of 8, which is 2.

Q: What are some examples of fractional indices?

Fractional indices combine roots and powers. For example, 125 to the power of one-third is the cube root of 125, which is 5. Also, 125 to the power of two-thirds equals 25. And 16 to the power of three-quarters equals 8.

Q: Do you find the root or the power first with fractional indices?

Find the root first, then apply the power. For 125 to the power of two-thirds, take the cube root of 125 to get 5, then square it to get 25. Working out the root first keeps the numbers smaller and easier to handle.

Key concept

Fractional indices write a root as a power. Raising 125 to the power 1/3 gives the cube root of 125, which is 5. The bottom of the fraction says which root to take, and the top says what power to raise it to.

The nth Root and Fractional Indices - introduction visual

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$The nth Root and Fractional Indices poster$

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Square, cube, and fourth root examples showing nth root calculations and corresponding powers

Cube root of 8 equals 2, showing nth root rule: multiplying n roots of a number returns the original value.

$Cube root of 8 equals 2, showing ∛8·∛8·∛8 = 8 and fractional indices rule a^(1/n) = ⁿ√a.$ $Cube root of 125 using fractional indices: 125 to the power ⅓ = ³√125 = 5, since 5³ = 125$ Fractional indices rule a^(m/n) with worked example 125^(2/3) = 25 using roots and powers

Fractional indices rule a^(m/n) with worked example 125^(2/3) = 25 using roots and powers

Fractional indices example: 16^(3/4) step by step, fourth root of 16 is 2, then 2^3 = 8

nᵗʰ Root

The nᵗʰ root means a number that is multiplied by itself $n$ times.
For example, the cube root of 27 is 3 because $3 \times 3 \times 3 = 27$ .

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What is $12 5^{\frac{1}{3}}$ ?

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What is the nth root of a number?

The nth root of a number is the value that, multiplied by itself n times, gives that number. For example, the cube root of 8 is 2, because 2 cubed equals 8. The small number in the root symbol tells you which root to take.

How do you find the nth root of a number?

To find the nth root, ask which number, raised to that power, gives the original number. For the cube root of 8, you look for the number that cubes to 8. That number is 2, because 2 times 2 times 2 equals 8.

What happens when you multiply an nth root by itself n times?

Multiplying the nth root of a number by itself n times returns the original number. For example, three cube roots of 8 multiplied together give 8. This is the first rule of roots, and it shows that a root reverses a power.

What does a power of one-third mean?

A power of one-third means the same as the cube root. Raising a number to the power of 1 over n is the same as taking its nth root. So 8 to the power of one-third equals the cube root of 8, which is 2.

What are some examples of fractional indices?

Fractional indices combine roots and powers. For example, 125 to the power of one-third is the cube root of 125, which is 5. Also, 125 to the power of two-thirds equals 25. And 16 to the power of three-quarters equals 8.

Do you find the root or the power first with fractional indices?

Find the root first, then apply the power. For 125 to the power of two-thirds, take the cube root of 125 to get 5, then square it to get 25. Working out the root first keeps the numbers smaller and easier to handle.

Course Overview

The nth Root and Fractional Indices

Video Lesson

Flashcards

nᵗʰ Root

Rules of the nᵗʰ Root

Roots as Fractional Indices

Example: 12531​

Fractional Indices Rule

Example: 1643​