Introduction To Powers And Roots
I. Introduction
Remember when we learnt multiplication, we learnt that multiplication is nothing but repeated addition of the same number.
We will recollect this once more.
$4 \times 3 = 4 + 4 + 4 = 12$, likewise, $2 \times 6 = 2 + 2 + 2 + 2 + 2 + 2 = 12$.
Adding three $4$s is written as $4 \times 3$ and adding six $2$'s is written as $2 \times 6$.
What happens when we multiply the same number repeatedly. Let us say we multiply four $3$'s:
$3 \times 3 \times 3 \times 3= 81$
In mathematics this is written as $3^4 = 81$ and we read this as $\unicode{x201C}Three\ to\ the\ power\ four\unicode{x201D}$
In multiplication when we write $2 \times 6 = 12$, we call the number $2$ as
multiplicand
and $6$
as multiplier
.When we say $3^4 = 81$ we call the number $3$ as $base$ and $4$ as the $exponent$ or $power$.
In case of multiplication, we have an inverse operation called division.
If $4 \times 5 = 20$ then we know that $20 \div 5 = 4$
In case of powers as well, we have an inverse operation, this is called $root$.
If we saw before, that $3^4 = 81$, we can say that the $4^{th}$ root of $81$ is $3$. This is written as using the root symbol ($\sqrt{\phantom{0}}$) as:
$\sqrt[4]{81} = 3$. This is read as $Fourth\ root\ of\ 81$.
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When the number over the root symbol is missing the root value is $2$.
So, if we see $\sqrt{16}$, it actually means $\sqrt[2]{16}$
But, be careful, multiplication is commutative, so we can write $4 \times 5 = 5 \times 4 = 20$ and so, we can write:
$20 \div 4 = 5$ or $20 \div 5 = 4$.
But, power, is not commutative, so:
$3^4 \ne 4^3$
and $\sqrt[3]{81} \ne 4$
II. Definitions
$\underline{\text{Names Used For Common Exponents And Roots}}$
When the power or the root is either $2$ or $3$, they are called by commonly used names.
When the root or exponent is $2$ we call it as $square$.
When the root or exponent is $3$ we call it as $cube$.
So, $p^2$ is read as $\unicode{x201C}p\ squared\unicode{x201D}$ and $p^3$ is read as $\unicode{x201C}p\ cubed\unicode{x201D}$.
Similarly,
$\sqrt{p}$ (or $\sqrt[2]{p}$) is read as $\unicode{0x201C} square\ root\ of\ p \unicode{0x201D}$ and $\sqrt[3]{p}$ is called $\unicode{0x201C} cube\ root\ of\ p \unicode{0x201D}$
III. Calculations
$\underline{\text{Calculating Powers}}$
Powers can be calculated in a simple manner by multiplying the base as many times as the value of the power.
For example:
$5^3 = 5 \times 5 \times 5 = 125$
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$\underline{\text{Calculating Roots}}$
Let us try to understand roots a little better, before we get calculating roots. When we say $\sqrt[4]{625}$ , we are trying to find a number, which, when multiplied with itself $4$ times, will give a result of $625$. If the number is $N$, then
$N \times N \times N \times N$ should give us $625$.
Factorizing $625$, we get
$625 = 5 \times 5 \times 5 \times 5$
Here we observe that if we multiply $5$ four times, we get $625$.
$\texttip{\therefore}{therefore} \sqrt[4]{625} = 5$.
Let us try to find another root, let us say $\sqrt[3]{216}$
Factorizing $216$, we get:
$216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$
We know that we are trying to find out the cube $(3)$ root of $216$, so let us try to re-arrange the factors into $3$ equal groups:
$= 2 \times 3 \times 2 \times 3 \times 2 \times 3$
$= (2 \times 3) \times (2 \times 3) \times (2 \times 3)$
Now we see that if we multiply $(2 \times 3)$ thirce we get $216$.
$\texttip{\therefore}{therefore} \sqrt[3]{216} = 2 \times 3 = 6$
We will find out the value of $\sqrt{1089}$. (Remember $\sqrt{1089}$ is same as $\sqrt[2]{1089}$)
Factorizing $1089$ we get:
$1089 = 3 \times 3 \times 11 \times 11$
re-arranging the factors into two equal groups we get:
$1089 = (3 \times 11) \times (3 \times 11)$
$\texttip{\therefore}{therefore} \sqrt{1089} = (3 \times 11) = 33$