Primary Mathematics/Powers, roots, and exponents

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Primary Mathematics
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Exponents[edit]

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,

x^n = \underbrace{x \cdot x \cdot x \cdot \,\,\cdots\,\, \cdot x}_{n factors}

where there are n x's to be multiplied.

Squaring numbers[edit]

A convenient way to note 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 5×5 or 25 square units. A nice way to write "5 squared" is 52, where 5 is called the base and 2 is called the exponent.

5x5.PNG

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

Table of perfect squares[edit]

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
212 441
222 484
232 529
242 576
252 625
262 676
272 729
282 784
292 841
302 900

Cubing numbers[edit]

Similarly, a convenient way to note 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 5×5×5 or 125 cubic units. A nice way to write "5 cubed" is 53, where 5 is called the base and 3 is called the exponent.

5cube.svg

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

Table of perfect cube numbers[edit]

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[edit]

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[edit]

A table of xn.
Bases are on the left, exponents are across the top:
x1/6 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 19683 59049 3n
4 1 4 16 64 256 1024 4096 16384 65536 262144 1048576 4n
5 1 5 25 125 625 3125 15625 78125 390625 1953125 9765625 5n
6 1 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 6n
7 1 7 49 343 2401 16807 117649 823543 5764801 40353607 282475249 7n
8 1 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 8n
9 1 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 9n
10 1 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000 10n
11 1 11 121 1331 14641 161051 1771561 19487171 214358881 2357947691 25937424601 11n
12 1 12 144 1728 20736 248832 2985984 35831808 429981696 5159780352 61917364224 12n
13 1 13 169 2197 28561 371293 4826809 62748517 815730721 10604499373 137858491849 13n
14 1 14 196 2744 38416 537824 7529536 105413504 1475789056 20661046784 289254654976 14n
15 1 15 225 3375 50625 759375 11390625 170859375 2562890625 38443359375 576650390625 15n
16 1 16 256 4096 65536 1048576 16777216 268435456 4294967296 68719476736 1099511627776 16n ...
x 1 x x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

Note: 00 is undefined.

Properties of Exponents[edit]

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


\begin{align}

x^mx^n & = x^{m+n}\\

\dfrac{x^m}{x^n} & = x^{m-n}\\

(xy)^n & = x^ny^n\\

\left(\dfrac{x}{y}\right)^n & = \dfrac{x^n}{y^n}\\

(x^m)^n & = x^{mn}

\end{align}

The first, third, and fifth properties may be extended to several factors, as follows:


\begin{align}

x^{m_1}x^{m_2}x^{m_3} \cdots x^{m_n} & = x^{m_1+m_2+m_3+ \, \cdots \, +m_n}\\

(x_1x_2x_3 \cdots x_n)^m & = x_1^mx_2^mx_3^m \cdots x_n^m\\

( \cdots (((x^{m_1})^{m_2})^{m_3}) \cdots )^{m_n} & = x^{m_1m_2m_3 \cdots \, m_n}

\end{align}

Bases and exponents of one and zero[edit]

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.

Negative and noninteger powers[edit]

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

Roots[edit]

Roots are the inverse operation of powers:

Square root[edit]

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:

\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[edit]

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[edit]

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:

\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[edit]

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

\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[edit]

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[edit]

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:

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

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

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

Decimal exponents[edit]

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

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

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

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[edit]

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

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

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

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

Fractions as bases[edit]

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

(5/6)^2 = 5^2/6^2 = 25/36 \;

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

(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:

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

Negative bases[edit]

Negative bases can be handled normally for integer powers:

(-5)^2 = (-5) \times (-5) = 25 \;

(-5)^3 = (-5) \times (-5) \times (-5) = -125 \;

(-5)^4 = (-5) \times (-5) \times (-5) \times (-5) = 625 \;

(-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  -5^2 and (-5)^2. The former means the negative of 5 times 5, whereas the latter
means -5 squared. In other words,

-5^2 = -1\times5^2 = -1\times(5\times5) = -1\times25 = -25

but

(-5)^2=(-5)\times(-5)=25


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

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

Even roots, however, have no real solution:

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

(-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[edit]

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, with one being the negative of the other. 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"). However, the mathematical definition of the square root of x squared is the absolute value of x. Thus, square roots equations do not have two answers but two numbers can square to equal the same rational number.


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