Theory

Powers and roots

Powers with integer, rational and real exponents and their properties, n-th roots with their domain conditions, rationalization, and the comparison of powers and of roots.

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Powers with natural and integer exponents
Properties of powers

Powers with natural and integer exponents

Power with natural and integer exponent

For $a \in R$ and $n$ natural with $n \geq 1$ , the power $a^{n}$ is the product of $n$ factors equal to $a$ :

a^{n} = n factors a \cdot a \dots a .

The definition is extended to integer exponents by setting, for $a \neq = 0$ ,

a^{0} = 1, a^{- n} = \frac{1}{a ^{n}} .

These definitions are the only ones that preserve the properties of powers: from $a^{n} \cdot a^{0} = a^{n + 0} = a^{n}$ it follows that $a^{0} = 1$ , and from $a^{n} \cdot a^{- n} = a^{0} = 1$ it follows that $a^{- n} = \frac{1}{a ^{n}}$ . For example $2^{- 3} = \frac{1}{8}$ and $(\frac{2}{3})^{- 1} = \frac{3}{2}$ .

Remark

With a negative base the sign of the power depends on the parity of the exponent: $(- 2)^{4} = 16$ and $(- 2)^{3} = - 8$ . The expression $- 2^{4}$ denotes the opposite of $2^{4}$ , that is, $- 16$ , because exponentiation precedes the change of sign.

Properties of powers

For $a, b > 0$ and real exponents $m, n$ (with negative bases they hold for integer exponents):

product of powers with the same base: $a^{m} \cdot a^{n} = a^{m + n}$ ; for example $2^{3} \cdot 2^{4} = 2^{7} = 128$ ;
quotient of powers with the same base: $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ ; for example $\frac{5 ^{6}}{5 ^{4}} = 5^{2} = 25$ ;
power of a power: $(a^{m})^{n} = a^{mn}$ ; for example $(3^{2})^{3} = 3^{6} = 729$ ;
product of powers with the same exponent: $a^{n} \cdot b^{n} = (ab)^{n}$ ; for example $2^{3} \cdot 5^{3} = 1 0^{3} = 1000$ ;
quotient of powers with the same exponent: $\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}$ ; for example $\frac{6 ^{2}}{3 ^{2}} = 2^{2} = 4$ .

The properties turn products and quotients of powers into a single power and are the tool with which expressions are simplified; in exercises they are used in both directions, also to rewrite a power in a more convenient form, such as $8^{10} = (2^{3})^{10} = 2^{30}$ .

Warning

The properties concern products and quotients. For the power of a sum the binomial expansion holds, and in general $(a + b)^{n}$ is different from $a^{n} + b^{n}$ : for example $(3 + 4)^{2} = 49$ , while $3^{2} + 4^{2} = 25$ . Likewise the sum $a^{m} + a^{n}$ is in general different from $a^{m + n}$ : $2^{2} + 2^{3} = 12$ , while $2^{5} = 32$ .

The theory continues with 4 more sections (n th roots, powers with rational and real exponents, rationalization…), plus all worked examples, the solutions and the PDF.

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