Exponents and Roots
Exponents are shorthand for repeated multiplication, just like multiplication is a shortened form of repeated addition.
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 2×10 = 20
a + a + a + a + a + a + a + a + a + a = a×10 = 10a
A much easier improvement, right? We do something similar in multiplication using exponents.
In mathematics, we use superscripts to represent the number of times the number is multiplied by itself. These superscripts are the exponents.
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 210 = 1024
a × a × a × a × a × a × a × a × a × a = a10
Exponents shorten writing out long strings of repeated multiplication.
For example, 3×2×2×3×3×2×2 = 2×2×2×2×3×3×3 = 2433.
Note: You might also see exponents written out this way: a^10. This is commonly used on computers because it's easier to type.
Now for the technical math stuff.
In the expression an, we are saying that a is being multiplied by itself n number of times. We call a the base, and n is the exponent. The expression an is called a power, and is read as, "a raised to the power of n" or "a to the nth power." In my above example of 210, 2 is the base, 10 is the exponent (the number of times 2 is multiplied by itself), and we read it as "2 raised to the 10th power." Because 210 = 1024, 1024 is a power of 2.
Some powers are special because they come up quite frequently. a2 can also be read as "a-squared," and a3 as "a-cubed." Also, a1 = a (which is pretty boring, but still important to know!).
Powers are included in the order of operations. (PEMDAS) - Parenthesis, Exponents, Division/Multiplication, Addition/Subtraction. Exponents are pretty high up on the list!
Example: What is the value of 32 + 54? First, evaluate 32 = 9 and 54 = 625. Then, add them together, and the result is 634.
Example: What is the value of -1 × 25? First, evaluate 25 = 32. Then, multiply -1 and 32, and you get a result of -32.
In some cases, you need to plug in values for variables to evaluate.
Example: Evaluate ab2 for a = 3 and b = 5. It’s very important to note that this means a × b2. Some students get confused and think that a and b need to be multiplied first, and then square the result. If we were to do that, the problem would be written as (ab)2 instead. According to the order of operations, first the b must be squared, and then the result multiplied by a. When we substitute the values, we get 3 × 52 = 3 × 25 = 75. The other way, (ab)2, would be (3×5)2 = (15)2 = 225. Pay extra attention to how the problem is written and follow the order of operations.
Things get a little tricky when you throw negatives into the mix.
For (-2)2, when we write out the multiplication we have (-2) × (-2) = 4. (Remember that when you multiply 2 negatives, you get a positive product, and when you multiply a positive and a negative you get a negative product.) What happens for (-2)3? You would get (-2) × (-2) × (-2) = 4 × (-2) = -8. What about (-2)4? (-2)5? You'll find that when the base is negative and the exponent is an even number, the result is positive. If the base is negative and the exponent is an odd number, the result is negative.
Why did I use parentheses around the -2? Couldn't you just write -23? -24?
Take for example (-2)4 versus -24. Using our definition of exponents, (-2)4 = 16. For -24, because there are no exponents around the (-2), we are actually saying the same thing as "the opposite of 24". (Just like -3 is the opposite of 3.) Because 24 = 16, then -24 ("the opposite of 24") = -16. Those parentheses become crucial to properly evaluating exponents with negatives! For this reason, I suggest using parentheses when plugging in values to evaluate to eliminate the confusion.
Negative and Zero Exponents
What about negative exponents? Can zero be an exponent?
First, yes, zero can be an exponent, and it's a little weird. When the exponent is zero (as in 20), the result is 1. Any base (except zero) with a zero exponent is equal to 1. 20 = 1, b0 = 1, 19298430 = 1. (00 is very bizarre and we say the result is indeterminate.)
What about negative exponents? If we looked at the negative as meaning "opposite", and the exponent means repeating a multiplication, then we can ask, "What is the opposite of multiplication?" Division! Having a negative exponent means how many times we divide one by that number.
Example: 5-1 = 1 ÷ 5 = 0.2
Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008
An easier way of evaluating 5-3 would be 5-3 = 1 ÷ (5×5×5) = 1/53 = 1/125 = 0.008. In general, a-n = 1/an. So for 4-2, we can say 4-2 = 1/42 = 1/16 = 0.0625.
When evaluating expressions with exponents, there are some important points to remember:
- Follow the order of operations and watch out for common traps.
- Remember that exponents are shorthand for repeated multiplication. 23 is not the same as 2×3! It means 2×2×2.
- Be careful evaluating exponents with negative bases. Use parenthesis when necessary to help you remember.
- Negative exponents are the same as repeated division of one by a number, or you can use the easy shortcut to evaluate them faster.
- Negative exponents don't make a number negative! 2-3 = 1/23 = 1/8 = 0.125, not -8!
Square Roots and Radicals
A square root is defined as a number which when multiplied by itself gives a real non-negative number called a square.
A square root is best defined using geometry where, considering a square (which is a four sided polygon whose sides are all equal), a square root is defined as the length of the diagonal of this square (a diagonal is a line drawn from one vertex/corner to the opposite vertex of the square).
A radical is a root of a number. A square root is a radical. Roots can be square roots, cube roots, fourth roots and so on.
A square root is commonly shown as
where √ is known as the radical sign and a2 is know as the radicand.
A square root of a number can also be represented as
and a radical as
where we say that in the above, we're finding the nth root of x. For more on the above notation, refer to section on exponents.
A radical can also be represented as
A square root is also represented as
A cube root as
A fourth root as
Every square has two square roots; one positive and the other negative. This is shown as:
which is written as
This can be proved in the following way. Consider a number, a
the latter is because a negative multiplied by a negative equals a positive.
And so it follows that
Thus it follows that any real positive number has two roots. But when talking about radicals
in other words, only refers to +x which is known as the principal square root. So despite having said above that
we usually only consider
especially if the radical sign is used.
But if the question asked is in the form
always give both the positive and negative roots, i.e.
Although any real positive number can be considered a square number and thus has a square root, we only consider numbers with whole number square roots as squares.
Properties of Square Roots and Radicals
Properties of square roots and radicals guide us on how to deal with roots when they appear in algebra.
- Exponents, Radicals, and Scientific Notation https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals
- Fractional (Rational) Exponents http://www.purplemath.com/modules/exponent5.htm
- Exponents & Roots: Definition and Examples http://study.com/academy/lesson/exponents-roots-definition-examples.html
- Powers, Roots and Radicals http://www.intmath.com/numbers/4-powers-roots-radicals.php
- Exponents and Roots - GRE http://www.onlinemathlearning.com/exponents-roots-gre.html
- Exponent Worksheets http://www.onlinemathlearning.com/math-worksheets.html#Exponents
- Many roots and exponents worksheets (Ctrl F or scroll down; listed under “Numbers”) http://www.commoncoresheets.com/Algebra.php
- Square Root Worksheets http://www.homeschoolmath.net/worksheets/square-roots.php
- Exponents Worksheets http://www.homeschoolmath.net/worksheets/exponents.php
- Intro to Rational Exponents https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/rational-exponents-intro/v/basic-fractional-exponents
- Understanding Square Roots https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/alg1-radicals/v/understanding-square-roots
- Math Antics: Exponents and Square Roots https://www.youtube.com/watch?v=C_iKTTI1E34