GMAT Quant: Arithmetic – Roots & Powers

Arithmetic: Roots and Powers

We have been studying roots and powers since our grade school, still it never seizes to amaze us with the complexity of problem it presents.

What are Powers?

Power is a way of indicating that a particular number is being multiplied by itself for some number of times. For example, if we express 36, then it is equivalent to 3  x 3 x 3 x 3 x 3 x 3 =  729.

What are Roots?

Inverse of power is root.

10

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Let’s solve a few questions and clear the basic concepts:

Example 1. If 22p+2 = 64, and, q = 2p+1 , Then find the value of q – p.

  1. 3
  2. 5
  3. 10
  4. 13
  5. 17

Solution: 22p+2 = 26

So, 2p + 2 = 6

So, p = 2

Hence, q = 2p+1

q = 23 = 8

So, p + q = 2 + 8 = 10

Example 2 – \(If \; \sqrt{3 – x} = \sqrt{x} + 3 \; Then, \; x^{2} = ?\)

Solution: \((\sqrt{3 – x})^{2} = (\sqrt{x} + 3)^{2}\)
\(3 – x = x + 3 (\sqrt{x}) + 3 (\sqrt{x}) + 9\)
\(3 – x = x + 2 \times 3 (\sqrt{x}) + 9\)
\(3 – x = x + 6 (\sqrt{x}) + 9\)
\(2x + 6 \sqrt{x} + 6 = 0\)
\(2 (x + 3 \sqrt x + 3)\)
It can be written as,
\(x + 3 \sqrt x + 3 = 0\)
\(- 3 \sqrt x = x + 3\)
\((- 3 \sqrt x)^{2} = (x + 3)^{2}\)
\(9x = x^{2} + 6x + 9\)
\(x^{2} -3x + 9 = 0\)
Therefore, \(x^{2} = -3x + 9\)

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