Primary mathematics/Powers, roots, and exponents

Exponents

Exponents, or powers, are a way of indicating that a quantity is to be multiplied by itself some number of times. In the the expression 25, 2 is called the base and 5 is called the exponent, or power. 25 is shorthand for "multiply five twos together": 25 = 2×2×2×2×2 = 32. Notice that the exponent tells us how many bases to multiply, not how many multiplications to perform. (In fact, the number of multiplications is one less than the number of bases.) 25 is read "two raised to the fifth power" or simply "two to the fifth."

In general,

${\displaystyle x^{n}=\underbrace {x\cdot x\cdot x\cdot \,\,\cdots \,\,\cdot x} _{nfactors}}$

where there are n x's to be multiplied.

It is better to say that exponent is 1 multiplied by x n times, ${\displaystyle x^{n}=1\cdot \underbrace {x\cdot x\cdot x\cdot \,\,\cdots \,\,\cdot x} _{nfactors}}$. This is permissible because factor 1 is multiplicative identity, adding it does not change the product yet enables the empty one, ${\displaystyle x^{0}=1}$.

The definition of exponent corresponds to equivalent definition of multiplication: ${\displaystyle x\times }$ n is nothing more than x added to 0 n times,

${\displaystyle x\cdot n=0+\underbrace {x+x+\cdots +x} _{nfactors}.}$

You can even convert one to another, using #Properties of Exponents, ${\displaystyle x^{n}=x\cdot x\cdot \ldots \cdot x=a^{\chi }\cdot a^{\chi }\cdot \cdots \cdot a^{\chi }=a^{\chi +\chi +\cdots +\chi }=a^{\chi \cdot n}.}$

Squaring numbers

A convenient way to say that a number is to be multiplied by itself (for example 5×5) is to say that the number is squared. To help visualize this, picture a square which is 5 units long and 5 units wide. This square will then have an area of 52 or 25 square units. 52 can be read "five raised to the second" but is most commonly read "five squared."

Area of five rows by five columns = 5 × 5 = 52 = 25.

Table of perfect squares

You may want to memorize some perfect square numbers:

 Square Result 02 0 12 1 22 4 32 9 42 16 52 25 62 36 72 49 82 64 92 81 102 100 112 121 122 144 132 169 142 196 152 225 162 256 172 289 182 324 192 361 202 400

Cubing numbers

Similarly, a convenient way to say that a number is to be multiplied by itself and then itself again (for example 5×5×5) is to say that the number is cubed. To help visualize this, picture a cube which is 5 units long, 5 units wide, and 5 units high. This cube will then have a volume of 53 or 125 cubic units. 53 can be read "five to the third power" but is commonly read "five cubed."

Volume of a cube with height, width, and length of five = 5 × 5 × 5 = 53 = 125.

Table of perfect cube numbers

You may want to memorize some perfect cube numbers:

 Cube Result 03 0 13 1 23 8 33 27 43 64 53 125 63 216 73 343 83 512 93 729 103 1000

Higher powers

Numbers higher than three may also be used as exponents, although there is no common term for numbers raised to a fourth power or higher. For example, 54 = 5×5×5×5 = 625.

Table of higher powers

A partial table of xn.
Bases are on the left, exponents are across the top:
xn 0 1 2 3 4 5 6 7 8 9 10 11 12 ... n
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 2n
3 1 3 9 27 81 243 729 2187 6561
4 1 4 16 64 256 1024 4096
5 1 5 25 125 625 3125
6 1 6 36 216 1296 7776
7 1 7 49 343 2401
8 1 8 64 512 4096
9 1 9 81 729 6561
10 1 10 100 1000 10000
11 1 11 121 1331
12 1 12 144 1728
13 1 13 169 2197
14 1 14 196 2744
15 1 15 225 3375
16 1 16 256 4096
...
x 1 x x2 x3

Note: 00 is undefined.

Properties of Exponents

There are several properties of exponents which are frequently used to manipulate and simplify algebraic and arithmetic expressions.

It follows immediately that

${\displaystyle x^{m}\cdot x^{n}=\underbrace {x\cdot x\cdots \cdot x} _{m\,factors}\cdot \underbrace {x\cdot x\cdots \cdot x} _{n\ factors}=\underbrace {x\cdot x\cdots \cdot x} _{m+n\ factors}=x^{m+n}}$

(1)

${\displaystyle (xy)^{n}=x^{n}y^{n}}$

(2)

and

${\displaystyle (x^{m})^{n}=x^{mn}}$

(3)

These can be extended to multiple factors

{\displaystyle {\begin{aligned}x^{m_{1}}x^{m_{2}}x^{m_{3}}\cdots x^{m_{n}}&=x^{m_{1}+m_{2}+m_{3}+\,\cdots \,+m_{n}}\\(x_{1}x_{2}x_{3}\cdots x_{n})^{m}&=x_{1}^{m}x_{2}^{m}x_{3}^{m}\cdots x_{n}^{m}\\(\cdots (((x^{m_{1}})^{m_{2}})^{m_{3}})\cdots )^{m_{n}}&=x^{m_{1}m_{2}m_{3}\cdots \,m_{n}}\end{aligned}}}

Alternatively, we can use first property to obtain

${\displaystyle x^{0}=x^{1-1}=x^{1}\cdot x^{-1}\implies x^{-1}=x^{0}/x^{1}=1/x}$

(3)

and use others to write down

${\displaystyle {\dfrac {x^{m}}{x^{n}}}=x^{m-n}\quad {\text{and}}\quad \left({\dfrac {x}{y}}\right)^{n}={\dfrac {x^{n}}{y^{n}}}}$

(4)

Bases and exponents of one and zero

Any number raised to an exponent of one equals itself. So, for example, 51 = 5.

Any non-zero number raised to an exponent of zero equals one. So, for example, 50 = 1.

Zero raised to any positive exponent is still zero. So, for example, 05 = 0.

One raised to any exponent is still one. So, for example, 15 = 1.

Zero raised to an exponent of zero is not defined.

Fractional powers

A base may also be raised to a fractional, or decimal power. This will be covered later in the lesson.

Roots

Roots are the inverse operation of powers (i.e. ${\displaystyle {\sqrt[{n}]{x^{n}}}=x}$ and from the properties of exponents we have ${\displaystyle {\sqrt[{n}]{x}}=x^{1/n}}$):

Square root

The inverse operation of squaring a number is taking the square root of that number. So, for example, the square root of 25 is the number which must be multiplied by itself to equal 25. In this case, the answer is 5. There are two types of notation used here:

${\displaystyle {\sqrt {25}}=25^{1/2}=5}$

Note, however, that most square roots don't yield integers, and many don't even produce rational numbers.

Manually finding a square root

One method for manually taking square roots is to repeatedly do long division. Let's take the square root of 10 in this example. We would start by estimating the answer. Since 32 = 9 and 42 = 16, we know the answer lies between 3 and 4. Furthermore, since 10 is only one away from 9, but is 6 away from 16, we could estimate that the answer is one-seventh of the way between 3 and 4. This won't give an exact answer, and a seventh is ugly to work with, so let's use a fifth, instead. This gives us 3 1/5 or 3.2 as a starting estimate.

Now do long division to divide 10 by 3.2. We get 3.125. The average of 3.2 and 3.125 is (3.2 + 3.125)/2 = 6.325/2 = 3.1625, so that will be our next estimate.

Now do long division to divide 10 by 3.1625. We get 3.162055... (we didn't really need to go more than one digit beyond the number of decimal places we started with). The average of 3.1625 and 3.1621 is 3.1623, so that will be our next estimate.

Now do long division to divide 10 by 3.1623. We get 3.162255...

So, this method can be repeated to get the desired level of accuracy. The actual square root of 10 is 3.16227766...

Note that calculators or computers are used for most square root calculations, but knowing how to manually calculate a square root can be quite useful when no calculator is available.

If you would like to try this method yourself, try finding the square root of 7.

Cube root

The inverse operation of cubing a number is taking the cube root of that number. So, for example, the cube root of 125 is the number which must be multiplied by itself and then multiplied by itself again to equal 125. In this case, the answer is 5. There are two types of notation used here:

${\displaystyle {\sqrt[{3}]{125}}=125^{1/3}=5}$

Note, however, that most cube roots don't yield integers, and many don't even produce rational numbers.

Higher roots

Numbers higher than three may also be used as roots, although there is no common term for fourth roots or higher. For example:

${\displaystyle {\sqrt[{4}]{625}}=625^{1/4}=5}$

Note, however, that most higher roots don't yield integers, and many don't even produce rational numbers.

Combining powers and roots

The unit fraction notation used for roots previously may have given you the idea that roots are really the same as powers, only with a unit fraction (one over some number) instead of an integer as the exponent. Thus, the fractional notation is actually preferred in higher mathematics, although the root symbol is still used occasionally, especially for square roots.

Fractions as exponents

Other (non-unit) fractions may also be used as exponents. In this case, the base number may be raised to the power of the numerator (top number in the fraction) then the denominator (bottom number) may be used to take the root. For example:

${\displaystyle 8^{2/3}={\sqrt[{3}]{8^{2}}}={\sqrt[{3}]{64}}=4}$

Alternatively, you can take the root first and then apply the power:

${\displaystyle 8^{2/3}=({\sqrt[{3}]{8}})^{2}=2^{2}=4}$

Decimal exponents

Any fractional exponent can also be expressed as a decimal exponent. For example, a square root may also be written as:

${\displaystyle 25^{1/2}=25^{0.5}=5\;}$

Also, decimals which can't be expressed as a fraction (irrational numbers) may be used as exponents:

${\displaystyle 5^{3.1415926...}=156.99...\;}$

Such problems aren't easy to solve by hand using basic math skills, but the answer can be estimated manually. In this case, since 3.1415926 is between 3 and 4 (and considerable closer to 3), we know that the answer will be between 5^3 (or 125) and 5^4 (or 625), and considerable closer to 125.

Negative exponents

A negative exponent simply means you take the reciprocal (one over the number) of the base first, then apply the exponent:

${\displaystyle 25^{-0.5}=1/25^{0.5}=1/5\;}$

Alternatively, you can first apply the exponent (ignoring the sign), then take the reciprocal:

${\displaystyle 25^{-0.5}=5^{-1}=1/5\;}$

Fractions as bases

When a fraction is raised to an exponent, both the numerator and denominator are raised to that exponent:

${\displaystyle (5/6)^{2}=5^{2}/6^{2}=25/36\;}$

Fractions may also be used for both the base and exponent:

${\displaystyle (25/36)^{1/2}=25^{1/2}/36^{1/2}=5/6\;}$

In addition, negative fractional exponents may be used, taking the reciprocal of the base, as always, to find the solution.

Negative bases

Negative bases can be handled normally for integer powers:

${\displaystyle (-5)^{2}=(-5)\times (-5)=25\;}$

${\displaystyle (-5)^{3}=(-5)\times (-5)\times (-5)=-125\;}$

${\displaystyle (-5)^{4}=(-5)\times (-5)\times (-5)\times (-5)=625\;}$

${\displaystyle (-5)^{5}=(-5)\times (-5)\times (-5)\times (-5)\times (-5)=-3125\;}$

Note that negative bases raised to even powers produce positive results, while negative bases raised to odd powers produce negative results.

Be careful with negative signs. Since -5 = -1×5, there is a difference between ${\displaystyle -5^{2}}$ and ${\displaystyle (-5)^{2}}$. The former means the negative of 5 times 5, whereas the latter
means -5 squared. In other words,

${\displaystyle -5^{2}=-1\times 5^{2}=-1\times (5\times 5)=-1\times 25=-25}$

but

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

Roots and fractional/decimal powers are a bit trickier. Odd roots work out fine:

${\displaystyle (-125)^{1/3}=-5\;}$

Even roots, however, have no real solution:

${\displaystyle (-25)^{1/2}=?\;}$

${\displaystyle (-625)^{1/4}=?\;}$

Note that there is no real number, when multiplied by itself, which will produce -25, because 5×5 = 25 and -5×-5 = 25. There is actually a solution, called an imaginary number, but that won't be discussed until later lessons.

Principal root

Note that, since both 5×5 = 25 and -5×-5 = 25, when we are asked to take the square root of 25 there are, in fact, two valid answers, 5 and -5. Actually, any even root of a positive number will have two solutions, one positive and one negative. This may seem unusual, but, in higher mathematics, problems often have multiple solutions.

However, for many problems, only the positive value seems to physically work. For example, if we are asked to figure the length of the sides of a square yard which has an area of 25 square units, only 5 units on a side works. If we said "each side can also have a length of -5 units", that doesn't make any sense. For this reason, the positive solution is called the principal root, and, depending on the question, may be the only desired answer. In cases where either answer is valid, it is sometimes written as ±5 (read as "plus or minus five").