Power and root

Powers and roots are two sides of the same coin.

A power describes repeated multiplication, while a root is the inverse operation, where you find the number that must be multiplied by itself a certain number of times to give a result.

Both concepts appear in many areas of mathematics – from simple area calculations to large numbers in scientific contexts.

 

 

Powers

A power is a way of writing repeated multiplication. Instead of multiplying a number by itself many times, we use a small exponent above the number. It is written like this:

 

$$ \large a^n = a \cdot a \cdot a \cdot a \cdots a \ \ (n\ times) $$

 

Example:

 

$$ \large 3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81 $$

 

 

Special cases

There are some important and useful rules for the exponent:

 

  • Zero power: For all numbers \(a \ne 0\), it holds that \(a^0 = 1\).
  • First power: Here, \(a^1 = a\).
  • Negative powers: A negative exponent means taking the reciprocal value. For example: \(a^{-2} = \tfrac{1}{a^2}\).
  • Fractions as exponents: A fractional exponent connects powers and roots. For example: \(a^{\tfrac{1}{2}} = \sqrt{a}\) and \(a^{\tfrac{1}{3}} = \sqrt[3]{a}\).

 

 

Roots

Roots are the inverse operation of powers. If we know that \(9^2 = 81\), then the square root of 81 is 9.

 

$$ \large \sqrt{81} = 9 $$

 

In the same way, you can take cube roots (third root), fourth root, and so on:

 

$$ \large \sqrt[3]{729} = 9 \quad \text{because } 9^3 = 729 $$

$$ \large \sqrt[4]{6561} = 9 \quad \text{because } 9^4 = 6561 $$

 

In general, we can say that the n-th root of a number \(a\) is the number which, multiplied by itself \(n\) times, gives \(a\).

 

 

Connection between powers and roots

There is a close connection between powers and roots. In fact, all roots can be written as powers with fractional exponents:

 

$$ \large \sqrt[n]{a} = a^{\tfrac{1}{n}} $$

$$ \large \sqrt[n]{a^m} = a^{\tfrac{m}{n}} $$

 

 

Applications

Powers and roots are used throughout mathematics and science. Some examples:

 

  • Area and side lengths: If the area of a square is 49, you can find the side length by taking the square root: \( \large \sqrt{49} = 7\).
  • Pythagoras’ theorem: In a right-angled triangle, \( \large a^2 + b^2 = c^2\). If the legs \( \large a\) and \( \large b\) are known, the hypotenuse \( \large c\) can be found using a square root: \( \large c = \sqrt{a^2 + b^2}\).
  • Scientific notation: Large or small numbers are often written as powers of 10, e.g. \( \large 3.2 \cdot 10^5\).

 

 

Formulas

Power

$$ a^{-n} = {1 \over a^n} $$

$$ a^n \cdot a^p = a^{n+p} $$

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

$$ {a^n \over a^p} = a^{n-p} $$

$$ {a^n \over b^n} = \bigl( {a \over b} \bigr)^n $$

$$ (a^n)^p = a^{n\ \cdot \ p} $$

$$ a^{\frac{1}{2}}=\sqrt{a} $$

$$ a^{\frac{1}{3}}=\sqrt[3]{a} $$

$$ \sqrt[r]{a^p}=a^{\frac{p}{r}} $$

$$ 2a^2 = 2 \cdot a \cdot a $$

$$ (2a)^2 = (2a) \cdot (2a) = 4a^2 $$

Root

$$ \sqrt {a \cdot b}\ =\ \sqrt{a} \cdot \sqrt{b} $$

$$ \sqrt {a \over b}\ =\ {\sqrt{a} \over \sqrt{b} } $$

$$ \sqrt[m]{a}\cdot \sqrt[n]{a}=\sqrt[m \cdot n]{a^{m+n}} $$

$$ \sqrt[m]{a} \cdot \sqrt[m]{b}= \sqrt[m]{a\ \cdot \ b} $$

$$ \frac{\sqrt[m]{a}}{\sqrt[n]{a}}=\sqrt[m \cdot n]{a^{n-m}} $$

$$ \sqrt[n]{a}=a^{\frac{1}{n}} $$

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

$$ (\sqrt[n]{a})^m=a^{\frac{m}{n}} $$

$$ \frac{\sqrt[m]{a}}{\sqrt[m]{b}}=\sqrt[m]{\frac{a}{b}} $$

$$ \sqrt[n]{\sqrt[m]{a}}=\sqrt[m\ \cdot \ n]{a} $$