Indices

Every rule and operation relating to Exponents and Roots which can be tested on the GMAT has been listed here. Just make yourself comfortable with the operations and notations. Try to do the problems on your own before you look into the explanations. Enjoy!

Exponents

a^n = \underbrace{a \times \cdots \times a}_n

\frac{x^n}{x^m} = x^{n - m}

1 = \frac{x^n}{x^n} = x^{n - n} = x^0

  • Any number to the power 1 is itself.
  • Any nonzero number to the power 0 is 1

Negative integer exponents

a^{-1} = \frac{1}{a}

a^{-n} = \frac{1}{a^n}

a^{-n} \, a^{n} = a^{-n\,+\,n} = a^0 = 1

Exponentiation to a negative integer power can alternatively be seen as repeated division of 1 by the base. For instance,

3^{-4} = (((1/3)/3)/3)/3 = \frac{1}{81} = \frac{1}{3^{4}}

Identities and properties

a^{m + n} = a^m \cdot a^n

a^{m - n} =\frac{a^m}{a^n}

for a\ne0, and

(a^m)^n = a^{m \cdot n}

(a \cdot b)^n = a^n \cdot b^n

Exponentiation is not commutative: 23 = 8, but 32 = 9.

Similarly, exponentiation is not associative either: 2^3 to the 4th power is 8^4 or 4096, but 2 to the 3^4 power is 2^81 or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, the order is usually understood to be top-down, not bottom-up:

a^{b^c}=a^{(b^c)}\ne (a^b)^c=a^{(b\cdot c)}=a^{b\cdot c}.

Important Powers

  • The integer powers of one are one: 1n = 1.
  • If the exponent is positive, the power of zero is zero: 0n = 0, where n > 0.
  • If the exponent is negative, the power of zero (0^n, where n < 0) is undefined, because division by zero is implied.
  • If the exponent is zero, some authors define 0^0=1, whereas others leave it undefined, as discussed below.
  • If n is an even integer, then (-1)^n = 1
  • If n is an odd integer, then (-1)^n = -1

Surds

n^{th} root of a number a is a number b such that when n copies of b are multiplied together, the result is a. Note that if n is even, any negative number will not have a real nth root.

Fundamental operations

\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b} \qquad a \ge 0, b \ge 0 \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \qquad a \ge 0, b > 0

\sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m = \left(a^{\frac{1}{n}}\right)^m = a^{\frac{m}{n}},

where a and b are positive.

a^m a^n = a^{m+n}

\left({\frac{a}{b}}\right)^m = \frac{a^m}{b^m}

(a^m)^n = a^{mn}

For example:

\sqrt[3]{a^5}\sqrt[5]{a^4} = a^\frac{5}{3} a^\frac{4}{5} = a^\frac{25 + 12}{15} = a^\frac{37}{15}

\frac{\sqrt{a}}{\sqrt[4]{a}} = a^\frac{1}{2}a^\frac{-1}{4}= a^\frac{4 - 2}{8} = a^\frac{2}{8} = a^\frac{1}{4}

If you are going to do addition or subtraction, then you should notice that the following concept is important.

\sqrt[3]{a^5} = \sqrt[3]{aaaaa} = \sqrt[3]{a^3a^2} = a\sqrt[3]{a^2}

To simplify, addition and subtraction is a matter of “grouping like terms”. For example,

\sqrt[3]{a^5}+\sqrt[3]{a^8}

=\sqrt[3]{a^3a^2}+\sqrt[3]{a^6 a^2}

=a\sqrt[3]{a^2}+a^2\sqrt[3]{a^2}

=({a+a^2})\sqrt[3]{a^2}

Working with surds

a\sqrt{b}+c\sqrt{b} = (a+c)\sqrt{b}

\sqrt{a^2 b} = a \sqrt{b}

The above can be combined with index reduction: \sqrt[6]{a^6b^4} = \sqrt[3\cdot 2]{a^2a^2a^2b^2b^2} = \sqrt[3]{a^3b^2} = a\sqrt[3]{b^2}

\sqrt[n]{a^m b} = a^{\frac{m}{n}}\sqrt[n]{b}

\sqrt{a} \sqrt{b} = \sqrt{ab}

\frac{\sqrt{a}}{\sqrt{b}} = \sqrt\frac{a}{b}

\left(\frac{a}{\sqrt{b}}\right)\left(\frac{\sqrt{b}}{\sqrt{b}}\right) = \frac{{a}\sqrt{b}}{b}(\sqrt{a}+\sqrt{b})^{-1} = \frac{1}{(\sqrt{a}+\sqrt{b})} = \frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})} = \frac{\sqrt{a}- \sqrt{b}} {a - b}

The last of these may serve to rationalize the denominator of an expression, moving surds from the denominator to the numerator. It follows from the identity

(\sqrt{a}+\sqrt{b})(\sqrt{a}- \sqrt{b}) = a - b

which exemplifies a case of the difference of two squares. Variants for cube and other roots exist, as do more general formulae based on finite geometric series.

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