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Powers and roots

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Powers and roots#

“I have the Power!” - He-Man (1983)

If a number is multiplied by itself, we can express this with an exponent. For instance

\(3\cdot3=3^2\)

\(3\cdot3\cdot3=3^3\)

\(3\cdot3\cdot3\cdot3=3^4\)

\(3\cdot3\cdot3\cdot...\cdot3=3^n{,}\ n\in\mathbb{Z}\)

Here 3 is called the base, the number which is being multiplied, and the number of times the base is being multiplied is called the exponent. The base can also be a real variable, for instance

\(x\cdot x\cdot x\cdot...\cdot x=x^n{,}\ x\in\mathbb{R}{,}\ n\in\mathbb{Z}\)

Since the exponent can be any integer, we can define

\(x^0=1{,}\ x\in\mathbb{\mathbb{R}}\) and

\(x^{-n}=\frac{1}{x^n}{,}\ x\in\mathbb{R}{,}\ n\in\mathbb{Z}^+\)

Examples#

a) \(5\cdot5\cdot5=5^3\)

b) \(10\cdot10\cdot10\cdot10\cdot10\cdot10=10^6\)

c) \(y\cdot y\cdot y\cdot y=y^4\)

d) \(9^0=1\)

e) \(3\cdot\left(4x^2\right)^0=3\cdot1=3\)

f) \(1^{-2}=\frac{1}{2}\)

g) \(x^{-4}=\frac{1}{x^4}\)

Here are the laws of power expressions, which apply to all real numbers as bases \(x{,}\ y\in\mathbb{R}\) and integers as exponents \(a{,}\ b\in\mathbb{Z}\).

  • Multiplying with the same base \(x^a\cdot x^b=x^{a+b}\)

  • Dividing with the same base \(\frac{x^a}{x^b}=x^{a-b}\)

  • Power of a product \(\left(xy\right)^a=x^ay^a\)

  • Power of a quotient \(\left(\frac{x}{y}\right)^a=\frac{x^a}{y^a}{,}\ y\ne0\)

  • Power of a power \(\left(x^m\right)^n=x^{mn}\)

Roots can be expressed as powers, when exponents are fractions

  • \(\sqrt[m]{x}=x^{\frac{1}{m}}{,}\ x\ge0\)

and we get the following

  • \(\sqrt{x}\sqrt{y}=\sqrt{xy}\), since \(\sqrt{x}\sqrt{y}=x^{\frac{1}{2}}\cdot y^{\frac{1}{2}}=\left(xy\right)^{\frac{1}{2}}=\sqrt{xy}\)

  • \(\sqrt[a]{x}\sqrt[a]{y}=\sqrt[a]{xy}\), since \(\sqrt[a]{x}\sqrt[a]{y}=x^{\frac{1}{a}}\cdot y^{\frac{1}{a}}=\left(xy\right)^{\frac{1}{a}}=\sqrt[a]{xy}\)

  • \(\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}\), since \(\frac{\sqrt{x}}{\sqrt{y}}=\frac{x^{\frac{1}{2}}}{y^{\frac{1}{2}}}=\left(\frac{x}{y}\right)^{\frac{1}{2}}=\sqrt{\frac{x}{y}}\)

  • \(\frac{\sqrt[a]{x}}{\sqrt[a]{y}}=\sqrt[a]{\frac{x}{y}}\), since \(\frac{\sqrt[a]{x}}{\sqrt[a]{y}}=\frac{x^{\frac{1}{a}}}{y^{\frac{1}{a}}}=\left(\frac{x}{y}\right)^{\frac{1}{a}}=\sqrt[a]{\frac{x}{y}}\)

And last, we have exponent as a fraction

  • \(x^{\frac{m}{n}}=\sqrt[n]{x^m}=\left(\sqrt[n]{x}\right)^m\)

Examples#

a) \(x^3\cdot x^2=x^{3+2}=x^5\)

b) \(\frac{x^4}{x^3}=x^{4-3}=x^1=x\)

c) \(\left(3x\right)^3=3^3x^3=27x^3\)

d) \(\left(\frac{x}{y^2}\right)^2=\frac{x^2}{\left(y^2\right)^2}=\frac{x^2}{y^{2\cdot2}}=\frac{x^2}{y^4}\)

e) \(\left(4x\right)^{-2}=\frac{1}{\left(4x\right)^2}=\frac{1}{4^2x^2}=\frac{1}{16x^2}\)

f) \(64^{\frac{1}{3}}=\sqrt[3]{64}=4\)

g) \(3^{\frac{2}{5}}=\sqrt[5]{3^2}=\sqrt[5]{9}\)

h) \(\sqrt[3]{3}\sqrt[3]{9}=\sqrt[3]{3\cdot9}=\sqrt[3]{27}=3\)

i) \(\sqrt{x}\sqrt{x}=x^{\frac{1}{2}}\cdot x^{\frac{1}{2}}=x^{\frac{1}{2}+\frac{1}{2}}=x^1=x\)

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