GRE Algebra | Rules of Exponent

In algebraic expression x^{m}, x is the **base** and m is the **exponent**. For all positive number of x except x=1, if an equation contain x^{m} = x^{n} then it will be only possible when m = n.

**Here are basic rules of Exponents:**

- If a number raised to the power zero then it should be equal to 1.
x

^{0}= 1**Example:**2

^{0}= 1 - A negative exponent is the same as the reciprocal of the positive exponent.
x

^{-m}= 1/x^{m}**Example:**2

^{-4}= 1/2^{4}= 1/16 - If two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents.
(x

^{m}) (x^{n}) = x^{m+n}**Example:**(2

^{3})(2^{4}) = 2^{7}= 128 - If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents.
x

^{m}/x^{n}= x^{m-n}= 1/x^{n-m}**Example:**3

^{4}/3^{2}= 3^{2}= 9 - If two powers have different base but same exponent then we multiply the base of powers and exponent will remain same.
(x

^{m})(y^{m}) = (xy)^{m}**Example:**3

^{2}4^{2}= 12^{2}= 144 - If base is a fraction then the exponent of the power multiply with numerator and denominator separately.
(x/y)

^{m}= x^{m}/y^{m}**Example:**(2/3)

^{2}= 2^{2}/3^{2}= 4/9 - If power has an exponent then both the exponents multiplied.
(x

^{m})^{n}= x^{mn}**Example:**(3

^{2})^{3}= 3^{6}= 729

**Avoid the common mistakes like below:**

x

^{m}y^{n}≠ (xy)^{m+n}Here bases are not same so we cannot add the exponents.

(x

^{m})^{n}≠ x^{m}x^{n}Here exponents should be multiplied not added according to rule.

(x + y)

^{m}≠ x^{m}+ y^{m}Have a look at (x + y)

^{2}= x^{2}+ 2xy + y^{2}

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