Roots, Powers, & Exponents

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Exponentiation is a mathematical operation, written aⁿ, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication:

$a^n = \underbrace{a \times \cdots \times a}_n,$

just as multiplication by a whole number corresponds to repeated addition:

$a \times n = \underbrace{a + \cdots + a}_n.$

The exponent is usually shown as a superscript to the right of the base. The exponentiation aⁿ can be read as: a raised to the n-th power or a raised to the power [of] n, or more briefly: a to the n-th power or a to the power [of] n, or even more briefly: a to the n. Some exponents can be read in a certain way; for example a² is usually read as a squared and a³ as a cubed.

The power aⁿ can also be defined when the exponent n is a negative integer. When the base a is a positive real number, exponentiation is defined for real and even complex exponents n. The special exponential function e^x is fundamental for this definition. It enables the functions of trigonometry to be expressed by exponentiation. However, when the base a is not a positive real number and the exponent n is not an integer, then aⁿ cannot be defined as a unique continuous function of a.

Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations.

Exponentiation is used pervasively in many other fields as well, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.

1 Exponentiation with integer exponents
2 Square Root
3 Principal square roots of the first 20 positive integers
- 3.1 As decimal expansions

Exponentiation with integer exponents

The exponentiation operation with integer exponents only requires elementary algebra.

Positive integer exponents

a² = a·a is called the square of a because the area of a square with side-length a is a².

a³ = a·a·a is called the cube, because the volume of a cube with side-length a is a³.

So 3² is pronounced "three squared",and 2³ is "two cubed".

The exponent says how many copies of the base are multiplied together. For example, 3⁵ = 3·3·3·3·3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, and 243 is the power or, more specifically, the fifth power of 3 or 3 raised to the fifth power.

The word "raised" is usually omitted, and most often "power" as well, so 3⁵ is typically pronounced "three to the fifth" or "three to the five".

Formally, powers with positive integer exponents may be defined by the initial condition a¹ = a and the recurrence relation aⁿ⁺¹ = a·aⁿ.

Exponents one and zero

Notice that 3¹ is the product of only one 3, which is evidently 3.

Also note that 3⁵ = 3·3⁴. Also 3⁴ = 3·3³. Continuing this trend, we should have

3¹ = 3·3⁰.

Another way of saying this is that when n, m, and n - m are positive (and if x is not equal to zero), one can see by counting the number of occurrences of x that

$\frac{x^n}{x^m} = x^{n - m}.$

Extended to the case that n and m are equal, the equation would read

$1 = \frac{x^n}{x^n} = x^{n - n} = x^0$

since both the numerator and the denominator are equal. Therefore we take this as the definition of x⁰.

Therefore we define 3⁰ = 1 so that the above equality holds. This leads to the following rule:

Any number to the power 1 is itself.
Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 0⁰ is discussed below.

Negative integer exponents

Raising a nonzero number to the −1 power produces its reciprocal.

$a^{-1} = \frac{1}{a}$

Thus:

$a^{-n} = (a^n)^{-1} = \frac{1}{a^n}$

Raising 0 to a negative power would imply division by 0, and so is undefined.

A negative integer exponent can also be seen as repeated division by the base. Thus $3^{-4} = (((1/3)/3)/3)/3 = \frac{1}{81} = \frac{1}{3^{4}}$ .

Identities and properties

The most important identity satisfied by integer exponentiation is:

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

This identity has the consequence:

$a^{m - n} =\frac{a^m}{a^n}$

for a ≠ 0, and

$(a^m)^n = a^{m \cdot n}$ .

Another basic identity is

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

While addition and multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2·3 = 6 = 3·2), exponentiation is not commutative: 2³ = 8, but 3² = 9.

Similarly, while addition and multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2·3)·4 = 24 = 2·(3·4), exponentiation is not associative either: 2³ to the 4th power is 8⁴ or 4096, but 2 to the 3⁴ power is 2⁸¹ or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, the order is usually understood to be from right to left:

$a^{b^c}=a^{(b^c)}\ne (a^b)^c=a^{(b\cdot c)}=a^{b\cdot c}$

Square Root

In mathematics, a square root (√) of a number x is a number r such that r² = x, or in words, a number r whose square (the result of multiplying the number by itself) is x. Every non-negative real number x has a unique non-negative square root, called the principal square root and denoted with a radical symbol as √x. For example, the principal square root of 9 is 3, denoted √9 = 3, because Template:mbox. If otherwise unqualified, "the square root" of a number refers to the principal square root: the square root of 2 is approximately 1.4142.

Square roots often arise when solving quadratic equations, or equations of the form ax² + bx + c = 0, due to the variable x being squared.

Every positive number x has two square roots. One of them is √x, which is positive, and the other −√x, which is negative. Together, these two roots are denoted ±√x. Square roots of negative numbers can be discussed within the framework of complex numbers. Square roots of objects other than numbers can also be defined.

Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers. For example, √2 cannot be written exactly as m/n, where n and m are integers. Nonetheless, it is exactly the length of the diagonal of a square with side length 1. This has been known since ancient times, with the discovery that √2 is irrational attributed to Hipparchus, a disciple of Pythagoras. (See square root of 2 for proofs of the irrationality of this number.)

Principal square roots of the first 20 positive integers

As decimal expansions

The square roots of the perfect squares (1, 4, 9, and 16) are integers. In all other cases, the square roots are irrational numbers, and therefore their decimal representations are non-repeating decimals.

$\sqrt {1}$	$=\,$	1
$\sqrt {2}$	$\approx$	1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 78462
$\sqrt {3}$	$\approx$	1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 16909
$\sqrt {4}$	$=\,$	2
$\sqrt {5}$	$\approx$	2.2360679774 9978969640 9173668731 2762354406 1835961152 5724270897 2454105209 25638
$\sqrt {6}$	$\approx$	2.4494897427 8317809819 7284074705 8913919659 4748065667 0128432692 5672509603 77457
$\sqrt {7}$	$\approx$	2.6457513110 6459059050 1615753639 2604257102 5918308245 0180368334 4592010688 23230
$\sqrt {8}$	$\approx$	2.8284271247 4619009760 3377448419 3961571393 4375075389 6146353359 4759814649 56924
$\sqrt {9}$	$=\,$	3
$\sqrt {10}$	$\approx$	3.1622776601 6837933199 8893544432 7185337195 5513932521 6826857504 8527925944 38639
$\sqrt {11}$	$\approx$	3.3166247903 5539984911 4932736670 6866839270 8854558935 3597058682 1461164846 42609
$\sqrt {12}$	$\approx$	3.4641016151 3775458705 4892683011 7447338856 1050762076 1256111613 9589038660 33818
$\sqrt {13}$	$\approx$	3.6055512754 6398929311 9221267470 4959462512 9657384524 6212710453 0562271669 48293
$\sqrt {14}$	$\approx$	3.7416573867 7394138558 3748732316 5493017560 1980777872 6946303745 4673200351 56307
$\sqrt {15}$	$\approx$	3.8729833462 0741688517 9265399782 3996108329 2170529159 0826587573 7661134830 91937
$\sqrt {16}$	$=\,$	4
$\sqrt {17}$	$\approx$	4.1231056256 1766054982 1409855974 0770251471 9922537362 0434398633 5730949543 46338
$\sqrt {18}$	$\approx$	4.2426406871 1928514640 5066172629 0942357090 1562613084 4219530039 2139721974 35386
$\sqrt {19}$	$\approx$	4.3588989435 4067355223 6981983859 6156591370 0392523244 4936890344 1381595573 28203
$\sqrt {20}$	$\approx$	4.4721359549 9957939281 8347337462 5524708812 3671922305 1448541794 4908210418 51276