Indices - MathsTips.com

Base and Index:

If a real number $a$ is multiplied $m$ times in succession (where $m$ is a positive integer) then the product so obtained is called the $m$ -th power of $a$ and is written as $a^{m}$ (read as, $a$ to the power $m$ ). Thus, $a^{m}=a \times a \times a \times$ ….to $m$ factors. Here, $a$ is called the base of $a^m$ and $m$ is called the index or exponent of $a^{m}$ . For example, $x^{4}=x \times x \times x \times x$ . Here, $x$ is the base and 4 is the exponent of $x^{4}$ .

Note: In particular, $a^{2}$ is called the square of $a$ and $a^{3}$ is called the cube of $a$

Root:

If $a$ and $x$ are two real numbers and $n$ is a positive integer such that $a^n=x$ , then $a$ is called the $n$ -th root of $x$ and is written as $a=\sqrt[n]{x}$ or $a=x^{\frac{1}{n}}$ .

Hence, it can be clearly seen that the $n$ -th root of $x$ ( $\sqrt[n]{x}$ ) is such a number which when multiplied multiplied by itself $n$ times, i.e. it is such a number whose $n$ -th power is equal to $x$ .

$\therefore (\sqrt[n]{x})^n=x$

In particular, if $a^2=x$ , then $a$ is called the second root or square root of $x$ and is written as $a=\sqrt[2]{x}$ or $a=x^{\frac{1}{2}}$ or simply, $a=\sqrt{x}$ .

If $a^3=x$ , then $a$ is called the third root or cube root of $x$ and it is written as $a=\sqrt[3]{x}$ or $a=x^{\frac{1}{3}}$ .

For example,

Square root of 25 is 5, i.e. $\sqrt[2]{25}=\sqrt{25}=25^{\frac{1}{2}}=5$

[ $\because 5^2=25$ ]

Cube root of 27 is 3, i.e. $\sqrt[3]{27}=27^{\frac{1}{3}}=3$

[ $\because 3^3=27$ ]

Sixth root of 64 is 2, i.e. $\sqrt[6]{64}=64^{\frac{1}{6}}=2$

[ $\because 2^6=64$ ]

Note:

1. $\because 5^{2}=25$

$\therefore \sqrt{25}=5$

Again,

$\because (-5)^{2}=25$

$\therefore \sqrt{25}=(-5)$

So, it is very clear that 5 and (-5) are both square roots of 25. As a result, when we try to find the square root of a positive number $x$ , we actually mean $\pm\sqrt{x}$ . Similarly, when we try to find the cube root of a positive number $x$ , there are 3 roots of out of which only one is positive. In general, we have $n$ roots when we try to find the $n$ -th root of a positive number $x$ of which only one root is positive.

For simplicity, when we want to find the square root or cube root or $n$ -th root of a real positive number we shall always mean only the positive real root. So $\sqrt[6]{64}=64^{\frac{1}{6}}=2$

2. If $x$ is a real negative number and:

(i) $n$ is an odd positive integer then there exists no psitive $n$ -th root of $x$ but we shall always get a real negative $n$ -th root of $x$ , say $a$ , such that $a^{n}=x$ . For example, if $x=-27$ and $n=3$ , then $(-27)^{\frac{1}{3}}=-3$ or, $(-3)^{3}=(-27)$

(ii) $n$ is an even positive integer then there exists no real number $a$ such that $a^{n}=x$ , i.e. in this case there is no real $n$ -th root of $x$ . For example $(-2)^{\frac{1}{6}}$ has no real value, say $a$ , such that $a^6=(-2)$