Exponents

What are exponents?

Exponents are a shorthand used for repeated multiplication. Remember that when you were first introduced to multiplication it was as a shorthand for repeated addition. For example, you learned that: 4 × 5 = 5 + 5 + 5 + 5. The expression "4 × " told us how many times we needed to add. Exponents are the same type of shorthand for multiplication. Exponents are written in superscript (that is, a smaller number written above) after a regular-sized number.

For example: 2³ = 2 × 2 × 2 = 8. The number in larger font is called the base. The number in superscript is called the exponent. The exponent tells us how many times the base is multiplied by itself. In this example, 2 is called the base and 3 is called the exponent.

The expression 2³ is read aloud as "2 raised to the third power", or simply "2 cubed".

Here are some other examples:

6 × 6 = 6² (This would read aloud as "six times six is six raised to the second power" or more simply "six times six is six squared".)
7 × 7 × 7 × 7 = 7⁴ (This would read aloud as "seven times seven times seven times seven equals seven raised to the fourth power". There are no alternate expression for raised to the fourth power. It is only the second and third powers that usually get abbreviated because they come up more often. When it is clear what is being talked about, people often drop the words "raised" and "power" and might simply say "seven to the fourth".)

Basic Rules of Exponents

Multiplying Powers with the Same Base

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

$a^{n}$ means that you have $a$ a factor of $n$ times. If you add $m$ more factors of $a$ then you have $n+m$ factors of $a$.

Dividing Powers with the Same Base

${\frac {a^{n}}{a^{m}}}={\frac {\overbrace {a\cdot a\cdot a\cdot ...\cdot a} ^{\text{n factors of a}}}{\underbrace {a\cdot a\cdot a\cdot ...\cdot a} _{\text{m factors of a}}}}=a^{n-m}$

In the same way that $a^{n}\cdot a^{m}=a^{n+m}$ because you are adding on factors of $a$, dividing is taking away factors of $a$. If you have m factors of a in the denominator, then you can cross out m factors from the numerator. If there were n factors in the numerator, now you have n-m factors in the numerator.

Raising a Power to a Power

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

If you think about an exponent as telling you that you have so many factors of the base, then $(a^{n})^{m}$ means that you have $m$ factors of $a^{n}$. So you have m groups of $a^{n}$ and each one of those has $n$ groups of $a$. So you have $m$ groups of $n$ groups of $a$. So you have $n\cdot m$ groups of a, or $a^{n\cdot m}$

Products raised to powers

$(ab)^{n}=a^{n}b^{n}$

You can multiply numbers in any order you please. Instead of multiplying together n factors equal to ab, you could multiply all of the a 's together, multiply all the b 's together, then finish by multiplying $a^{n}$ times $b^{n}$

Quotients raised to powers

$\left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}$

You can raise a fraction to a power by raising both the numerator and the denominator to the power.

Any Nonzero number Raised to the Zero Power Is One

$a\neq 0\implies a^{0}=1$

This all means that as long as the base is not zero, when you have an exponent of zero, the expression is always equal to 1.

Proof:
$\ {\frac {a^{n}}{a^{n}}}=a^{n-n}=a^{0}=1\ ,\ a\neq 0$

Note that $0^{0}$ is undefined.

Negative Exponents

A negative sign on an exponent means that you need to take the reciprocal of the base, then make the exponent positive.

$\left({\frac {a}{b}}\right)^{-n}=\left({\frac {b}{a}}\right)^{n}$

Examples of Basic Rules of Exponents

Multiplying Powers with the Same Base
Add the exponents:

$3^{4}\cdot 3^{2}=(3\cdot 3\cdot 3\cdot 3)\cdot (3\cdot 3)=3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3=3^{6}$

Dividing Powers with the Same Base
Subtract the exponents:

${\frac {5^{7}}{5^{4}}}={\frac {5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5}{5\cdot 5\cdot 5\cdot 5}}={\frac {5\cdot 5\cdot 5}{1}}=5^{3}$

Raising a Power to a Power
Multiply the exponents:

$(4^{3})^{2}=4^{3}\cdot 4^{3}=(4\cdot 4\cdot 4)\cdot (4\cdot 4\cdot 4)=4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4=4^{6}$

Any Nonzero number Raised to the Zero Power Is One

$2^{0}=1$

$109^{0}=1$

$-92^{0}=-1$

$(-92)^{0}=1$

Scientific Notation

Scientific notation makes use of exponents. It is often used for very large or very small numbers. It's easier to write $1,574,000,000,000,000$ as $1.574\cdot 10^{15}$. To convert from regular notation to scientific notation, find the leftmost non-zero digit. Count how many places away it is from the ones digit. This is the exponent for 10. If the digit was on the right of the ones digit, the exponent will be negative. If it was the ones digit, the exponent will be zero. Then, move the decimal place of the original number so that exactly one nonzero digit is on the left. Write down this new number and $\cdot 10^{\text{exponent}}$. You're done!

Roots

Roots are the inverse operation of powers:

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:

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

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:

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

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:

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

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:

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

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

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

Decimal exponents

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

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

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

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,

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

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

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.

Licensing

Content obtained and/or adapted from:

• Exponents, Wikibooks: Algebra under a CC BY-SA license
• Powers, roots, and exponents, Wikibooks: Primary Mathematics under a CC BY-SA license