Arithmetic – Roots

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Roots

• Nth root of a number x, where n is a positive integer, is a number r

which, when raised to the power n yields x

rn = x [reference: Wikipedia.org]

where r is nth root of x.

• m√n is also called as mth root of n.

• √n is called as square root of n.

• 3√n is called as cube root of n.

• m√n = n1/m

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Roots Formula

• Formula 1: m√abc = m√a * m√b * m√c

• E.g: 3√216 = 3√27 * 3√8 . Proof ? 3√216 = 6 3√27 = 3

3√8 = 2

• Hence 3√216 = 3√27 * 3√8

• Formula 2: m√(a:b) = m√a: m√b

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Roots Formula

• Formula 3: (m√X)n = m√(Xn)

• E.g. (3√8)2 = 3√(82)

• Formula 4: n√X = mn√Xm

• E.g. 3√27 = 6√729

• Formula 5: n√X = X1/m

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Example Problems

• Quantity A: √16 * √49

Quantity B: √729

Sol: A

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Example Problems

• Given 3√a4 = a2x/y what is the value of x/y?

Sol:2/3

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Example Problems

• Quantity A: ( 32 – 33 +34 - 35 )/((√5 - √7 )*(√5+ √7)) ?

Quantity B: 1

Sol: A

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Example Problems

• Quantity A: √5 + √6

Quantity B: √4 + √7

Sol: A

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Example Problems

• Given x+1√729 = 3 , what is the value of x ?

Sol: x = 5

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Example Problems

• Derive the value of a7/2 ?

Sol: a3√a

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