Exponents on the GMAT

by on September 2nd, 2010

An exponent refers to the number of times the base is a factor.  For example,  4^3= 4*4*4 = 64. For a term with a coefficient in front of a variable raised to an exponent, it’s important to remember that the exponent only affects the variable. Knowing the order of operations is helpful to avoid simple exponent mistakes. For 2x^2, first you would square x and then multiply the result by 2.

Any number to the ^2 power is referred to as being “squared.” Any number to the ^3 power is called being “cubed.”

When you multiply two terms with the same base, you can add the exponents: 2^5*2^3 = 2^{5+3} = 2^8

When you divide two terms with the same base, you can subtract the exponent of the numerator from the exponent of the denominator: 6^8 /  6^2 = 6^6

If two exponents are separated by a parenthesis, you can multiply them: (8^2)^5 = 8^10

On the GMAT look for ways to rewrite bases so they are the same.

9^7*3^x = 3^17

(3^2)^7 * 3^x = 3^17

3^14 * 3^x = 3^17

14 + x = 17

x = 3

There is no quick way of combining exponents when the bases are added. Don’t be fooled if you see something like 3^2 + 3^6 = ?. The answer is NOT 3^8. To solve, you must multiple out each term and then find the sum.

Any nonzero number raised to a power of zero is equal to 1.

3^0 = 1

However 0^0 = undefined.

A negative exponent is another way of expressing a fraction: x^{-1} = 1 / x^1

4^-2 = 1/4^2 = 1/16

A fractional exponent is another way of expressing a root: x^{1/n} = root{n}{x^1}

7^{2/3} = root{3}{7^2}

8^{1/3} = root{3}{8} = 2

When a fraction is raised to an exponent, you must distribute the power both to the numerator and the denominator:

(1/2)^3 = 1^3 / 2^3 = 1/8

Notice how the fraction will actually decrease in number as the exponent increases.

(1/2)^4 = 1^4 / 2^4 = 1/16

A negative number raised to an even exponent will always be positive. The negative sign will cancel itself out.

(-2)^2 = -2 * -2 = 4

However a negative numbers raised to an odd exponent will remain negative.

(-3)^3 = -3 * -3 * -3 = -27

Large numbers and very small decimals are often expressed with exponents using scientific notation. Scientific notation involves writing the number as a product of a decimal and the number 10 raised to a certain power.

The number of the exponent indicates the number of places the decimal moves.

10^7 = 1 + 7 zeros = 10,000,000

.036 * 10^4 = 360 (the decimal moves four places to the right)

.0000000857 * 10^6 = .0857

5.6 * 10^{-4} = .00056 (Since it’s a negative exponent, the decimal will move to the left.)

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