Re-Learn Math With Me

Yay! I have essentially graduated from my previous book, Basic Math & Pre-Algebra For Dummies. Now I will be reviewing from Schaum's Outlines Elementary Algebra. I am using their second edition, although I doubt it is for sale anymore. I have had this book for years. My mother bought it for me in the hopes that it would help me learn math...and now it is finally being used for its intended purpose! I'm sure whatever new edition they have to offer is just as good. The book is short on explanations and big on problems, which works well for my learning style. I learn by example.

For this post, I am also using my GMAT for Dummies book. I really like their section on bases and exponents. This is a great book for you to use if you are studying for the GMAT, like I am. I highly recommend it.

"As multiplication can be thought of as repeated addition, you can think of exponents as repeated multiplication; you multiply a number times itself" -GMAT for Dummies

Think about that statement; let's say I'm multiplying 3 x 4. What I am doing is adding 3 four times:
3+3+3+3 = 3 x 4 = 12

What happens if I multiplied 3 by itself four times?
3x3x3x3 = 81

My result from multiplying 3 is a far cry from adding 3 four times.

This is where exponents come in. They are a shorthand notation that lets us know that we are going to be multiplying a number by itself.

The above example of 3 multiplied by itself four times would be expressed as $3^{4}$

You might be asking, "What is the relationship between bases, exponents, and powers?"

I'm glad you asked.

The base is a repeated factor that a power will be applied to.

An exponent is the number that indicates how many times the base will be multiplied.

The power is another word for the exponent. The word "power" is used to describe what was done to the base with the exponent.

$3^{4} = 81$

3 is the base, 4 is the exponent, and 81 is the fourth power of 3.

What if we add a number ahead of the base? (let's say the base is unknown, for instance)

$3x^{4}$

The number 3, ahead of the new base, x, is called a coefficient. A coefficient is a number or symbol that is multiplied with a variable.

3 is the coefficient, x is the base, and 4 is the exponent. This reads as "3 times x to the 4th power." Remember that PEMDAS implies that you perform the operation of solving x to the 4th power first and then multiplying the result by 3.

Don't let the coefficient confuse you. Just remember that "The power governs only the number immediately below it!" -GMAT for Dummies

This means that only the number immediately to the left of the exponent (the base) will be affected. Unless the expression is in parentheses in which case you would turn to PEMDAS again:

$\left ( 3x^ \right )^{4}$

We would multiply the exponent 4 by both 3 and x because they are in parentheses, meaning (3x)(3x)(3x)(3x)= $81x^{4}$

There are some interesting properties when you are working with negative exponents:

When a negative number is raised to an odd exponent, the power stays negative:

- $3^{3}$ = (-3)(-3)(-3) = 9(-3) = -27

When a negative number is raised to an even exponent, the power becomes positive:

- $3^{4}$ = (-3)(-3)(-3)(-3) = (9)(9) = 81

As you can see by my examples, negative numbers are basically cancelled out when there are even exponents.

We can backtrack on this as well. If we have a power of -64, we can conclude that the base must have been -4, because the exponent (3) had to be odd to have resulted in a negative number.

Following the logic of even powers, anytime you have an exponent of 2, you have two possible roots: one positive and one negative (When we get to quadratic equations, you will understand this). Math has a funny way of coming together.

$5^{2}$ = (5)(5) = 25

$-5^{2}$ = (-5)(-5) = 25

Adding and Subtracting Exponents

You can only add or subtract like terms, meaning that the base and the exponent must be the same. The coefficient is the only number that will change.

Examples of like terms:

4 $a^{3}$ , $a^{3}$ , 10 $a^{3}$

Let's do an example of adding exponents:

4 $a^{3}$ + $b^{2}$ + $a^{3}$ + 5 $b^{2}$ + 10 $a^{3}$ + 6 $b^{2}$ = 15 $a^{3}$ + 12 $b^{2}$ <--We cannot simplify this equation any further unless we know of a relationship between a and b or we know what a and b are equal to.

Now an example of subtracting exponents:

10 $a^{3}$ - $b^{2}$ - $a^{3}$ - 5 $b^{2}$ - 4 $a^{3}$ - 6 $b^{2}$ = 5 $a^{3}$ - 12 $b^{2}$

This is going to be a slight repeat of my previous post on scientific notation but...

Multiplying and Dividing Exponents

Multiplication

If the bases are the same, add the exponents:

( $a^{3}$ )( $a^{5}$ ) = (a)(a)(a) x (a)(a)(a)(a)(a) = $a^{8}$

$a^{3} \times a^{-5} = a\cdot a\cdot a \left ( \frac{1}{a\cdot a\cdot a\cdot a\cdot a} \right ) = \frac{1}{a^{2}} = a^{-2}$

If the bases are different, but the exponents are the same:

$5^{3}$ x $2^{3}$ = (5)(5)(5)(2)(2)(2) = [(5)(2)][(5)(2)][(5)(2)] = (10)(10)(10) = $10^{3}$

$x^{4} \times y^{4} =\left ( xy \right )^{4}$

If the power is raised to another power, multiply the exponents:

$\left (t^{2} \right )^{3} = \left (t\cdot t \right )^{3} = \left (t\cdot t \right) \left (t\cdot t \right) \left (t\cdot t \right) = t\cdot t\cdot t\cdot t\cdot t\cdot t = t^6$ <--You might think it seems like we are adding t + t + t + t + t + t, but if were adding the t's, then we would have 6t, not t to the 6th power.

$\left ( 6^{4} \right )^{5}=6^{20}$

If you have an expression in parentheses that is raised to a power, use PEMDAS. Keep in mind that everything in parentheses is a base (there are no coefficients in parentheses):

$\left ( 7r \right )^{3} = 343r^{3}$

Division

If the bases are the same, subtract the exponents:

$a^{5}\div a^{3} = a\cdot a\cdot a\cdot a\cdot a \left ( \frac{1}{a\cdot a\cdot a} \right ) = a^{2}$

$a^{3} \div a^{5} = a^{-2}$

If the bases are different, but the exponents are the same:

$12^{4} \div 3^{4} = 4^{4}$

$\left ( xy \right )^{4} \div x^{4} = y^{4}$

For anything else, divide coefficients as usual (use the same rules as above for bases) and subtract exponents:

$8^{6} \div 4^{2} = \frac{\left ( 2^{3} \right )^{6}}{\left ( 2^{2} \right )^{2}} = 2^{18}\div 2^{4} = 2^{18}\left ( \frac{1}{2^{4}} \right ) = 2^{14}$ <---In this case, we do not have any coefficients. Only bases.

$8t^{6} \div 4t^{2} = \left ( \frac{8}{4} \right )\left ( \frac{t^{6}}{t^{2}} \right )= 2t^{4}$ <--- FYI we can perform the equation this way due to the associative property of multiplication. Keep in mind that 8 and 4 are coefficients in this example.

$20x^{2}y^{5}\div 4xyz = \left (\frac{20}{4} \right )\left ( \frac{x^{2}}{x} \right )\left ( \frac{y^{5}}{y} \right )\left ( \frac{1}{z} \right ) = \frac{5xy^{4}}{z} = 5xy^{4}z^{-1}$

Negative Exponents

Negative exponents are simply reciprocals of positive exponents.

$5^{-2} = \frac{1}{5^{2}} = \frac{1}{25}$

Keep in mind that a negative exponent does not make the base negative:

$3^{-3}\neq -27; -3^{3} = -27$

$3^{-3} \neq \frac{1}{-27}; 3^{-3}=\frac{1}{27}$

$3^{-3} = \left ( \frac{1}{3} \right )\left ( \frac{1}{3} \right )\left ( \frac{1}{3} \right ) = \left ( \frac{1}{27} \right ) = 27^{-1}$

$-3^{3} = \left ( -3 \right )\left ( -3 \right )\left ( -3 \right )=-27$

Fractional Exponents

Exponents can be written in fractional form. The top number is still the numerator and the bottom number is still the denominator. If you were to solve, say, $8^{2/3}$ , you would be raising 8 to its 2nd power and then finding the 3rd root.

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

(4)(4)(4) = 64

You can also find roots by using prime factorization:

   64
/ \
   8 8
/ \   / \
   2 4 2 4

We can see that we have (2)(4)(2)(4) = (2)(2)(4)(4) = (4)(4)(4)

Always raise the number by the power of the numerator and solve for the root using the denominator. Think of the example as:

$8^{\frac{2}{3}} = \left ( 8^{2} \right )^{\frac{1}{3}} or \left ( 8^{\frac{1}{3}} \right )^{2}$

Understanding the Power of 0 and 1

Always remember this:

A number to the power of 0 = 1
A number to the power of 1 = number

So, $5^{0}=1$ and $5^{1}=5$

Let's say we have $5^{3}$

$5^{3} \div 5^{3} = 5^{0}$

$5^{3} = 125$

$125\div 125 = 1$

$5^{3}\div 5^{2} = 5^{1}$

$5^{1}=5$

$125 \div 25= 5$

Below are some more rules to remember:

Every number raised to the power of 1 equals that number itself
Every number (except 0) raised to the power of 0 is equal to 1.
0⁰, is of course, equal to 0 (were you surprised?)
The number 0 raised to the power of any number (except 0) equals 0, because no matter how many times you multiply 0 by itself, the result is 0.
0⁰ is "undefined" (doesn't equal any number)
The number 1 raised to the power of any number equals 1, because no matter how many times you multiply 1 by itself, the result is 1.

Next up...roots and radicals

Re-Learn Math With Me

Bases, Exponents, and Powers (oh my!)

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