On this lesson, I will show you the basics of the mathematical expressions.
First of all, I will show you how to add and subtract "algebrically".
Let's have an example. I have 2 red, 3 green and 2 blue cars. How many care have I got totally?
The answer is simple. I have 2 + 3 + 2 = 7 cars totally.
Now, let's represent the cars with a variable.
Exercise 1: What is a variable?
So, we have the variable c.
We have 2 red cars (2c), 3 green (3c) and 2 blue (2c).
The total is \$2c + 3c + 2c = (2 + 3 + 2)c = 7c\$
As you can observe, the variables, like the numbers, can be added. Let's have some examples of variables:
Example 1: Find the sum: \$7a + 5a\$
Answer: We add normally. \$7a + 5a = (7 + 5)a = 12a\$
Exercise 2: Find the solutions:
a. \$25a + 36a =\$
b. \$41b + 23b =\$
c. \$199c + 201c =\$
d. \$25x - 5x =\$
e. \$42y - 20y - 2y =\$
Exercise 3: Find the solutions: (more challenging)
a. \$2t + 4t + 6t + 8t =\$
b. \$a + 3a + 5a =\$
c. \$5a - 6a =\$
d. \$25c - 3c =\$
Exercise 4: Prove that \$12a + 3a - 4b + 5b = 15a + b\$
Exercise 5: Find the solution of \$4a + 3a - 2b + b\$ for \$a = 3\$ and \$b = 2\$
Exercise 6: Find the soluction of \$15a + 2b - 4b + 3c + 2a - 3c + 2d\$ for \$a = 3\$, \$b = 4\$,
\$c = 5\$, \$d = 20\$
Challenge exerise: Find the solution of \$2a + 3b + 8a + 4c + 2c + 5b + 4c + 2b\$ if you know that
\$a + b + c = 20\$
(As you can see, variables represent all numbers, like \$1, 2, 3, 4, 0.75, \dfrac {1}{2}\$ etc.)
Now, you will learn about the exponentiation.
Let's have an example. Calculate the following:
a. \$2^3 =\$ (The cube of 2 is ...)
b. \$3^2 =\$ (The square of 3 is ...)
c. \$6^2 =\$ (The square of 6 is ...)
d. \$5^3 =\$ (The cube of 3 is ...)
e. \$8^2 =\$
Answer:
a. \$2^3 = 2 \cdot 2 \cdot 2 = 4 \cdot 2 = 8\$
b. \$3^2 = 3 \cdot 3 = 9\$
c. \$6^2 = 6 \cdot 6 = 36\$
Calculate the remaining expressions!
Exercise 7: Calculate!
a. \$5^3 =\$
b. \$4^4 =\$ (The fourth power of 4 is ...)
c. \$3^5 =\$ (The fifth power of 3 is ...)
d. \$2^6 =\$
e. \$12^2 =\$
f. \$20^2 =\$
g. \$15^2 =\$
h. \$(12 + 2)^2 =\$
j. \$(15 - 14)^{500} =\$
i. \$(20 - 7)^2 =\$
Observation:
As you can see, the exponent (the number on the upper right corner) tell us how much times we have to multiply the base (the big number).
Exercise 8: Calculate!
a. \$1^{500}\$
b. \$1^{280}\$
c. \$10^1 =\$
d. \$10^2 =\$
e. \$10^3 =\$
f. \$10^8 =\$
g. \$0^50 =\$
h. \$0^20 =\$
Observation:
As you can see from the exercise 8, we have find out that:
\$1^n = 1\$ (Remember: n is a variable. Variables represents all the numbers.)
\$0^n = 0\$
\$10^n = 1000...000\$ ( \$n\$ numbers)
Exercise 9: Write the following numbers as a power (there are several choices).
a. \$1000000 =\$
b. \$25 =\$
c. \$125 =\$
d. \$16 = \$
Exercise 10: Make a table with the squares of numbers from 1 to 10. I suggest you to learn it, because it's useful in plenty of exercises.
Exercise 11: Express in power:
a. \$a \cdot a \cdot a =\$
b. \$b \cdot b \ cdot b ... \cdot b =\$ (15 times)
c. \$2a \cdot 2a \cdot 2a =\$
d. \${(l^2)}^2 =\$
Exercise 12: Calculate the powers, like the example:
a. \$(2a)^2 = 2^2a^2 = 4a^2\$
b. \$(4b)^3 =\$
c. \$(2ab)^2 =\$
d. \${(ab)^2}^3\$
e. \$(2a + 3a)^4\$
Exercise 13: Calculate the following expressions:
a. \$(2a + 2b) + (3a + 3b) =\$
b. \$(5b + 4c) + (3a + 12) + (3c + 5 + a) + 2b\$
Remember: \$a + (b) = a + b\$
c. \$(2a + 3b) + (2a - 3b) =\$
d. \$(2a + 3b) - (2a + 3b) =\$
Remember: \$a - (b) = a - b\$
e. \$(2a + 3b) - (2a - 3b) =\$
Conclusion: As you can see, if a variable is on parenthesis, we following this rule:
If the operation before the parenthesis is plus, then we take out the parenthesis without any change.
If the operation before the parenthesis is minus, then we take out the parenthesis and we change the sign of the variables.
First of all, I will show you how to add and subtract "algebrically".
Let's have an example. I have 2 red, 3 green and 2 blue cars. How many care have I got totally?
The answer is simple. I have 2 + 3 + 2 = 7 cars totally.
Now, let's represent the cars with a variable.
Exercise 1: What is a variable?
So, we have the variable c.
We have 2 red cars (2c), 3 green (3c) and 2 blue (2c).
The total is \$2c + 3c + 2c = (2 + 3 + 2)c = 7c\$
As you can observe, the variables, like the numbers, can be added. Let's have some examples of variables:
Example 1: Find the sum: \$7a + 5a\$
Answer: We add normally. \$7a + 5a = (7 + 5)a = 12a\$
Exercise 2: Find the solutions:
a. \$25a + 36a =\$
b. \$41b + 23b =\$
c. \$199c + 201c =\$
d. \$25x - 5x =\$
e. \$42y - 20y - 2y =\$
Exercise 3: Find the solutions: (more challenging)
a. \$2t + 4t + 6t + 8t =\$
b. \$a + 3a + 5a =\$
c. \$5a - 6a =\$
d. \$25c - 3c =\$
Exercise 4: Prove that \$12a + 3a - 4b + 5b = 15a + b\$
Exercise 5: Find the solution of \$4a + 3a - 2b + b\$ for \$a = 3\$ and \$b = 2\$
Exercise 6: Find the soluction of \$15a + 2b - 4b + 3c + 2a - 3c + 2d\$ for \$a = 3\$, \$b = 4\$,
\$c = 5\$, \$d = 20\$
Challenge exerise: Find the solution of \$2a + 3b + 8a + 4c + 2c + 5b + 4c + 2b\$ if you know that
\$a + b + c = 20\$
(As you can see, variables represent all numbers, like \$1, 2, 3, 4, 0.75, \dfrac {1}{2}\$ etc.)
Now, you will learn about the exponentiation.
Let's have an example. Calculate the following:
a. \$2^3 =\$ (The cube of 2 is ...)
b. \$3^2 =\$ (The square of 3 is ...)
c. \$6^2 =\$ (The square of 6 is ...)
d. \$5^3 =\$ (The cube of 3 is ...)
e. \$8^2 =\$
Answer:
a. \$2^3 = 2 \cdot 2 \cdot 2 = 4 \cdot 2 = 8\$
b. \$3^2 = 3 \cdot 3 = 9\$
c. \$6^2 = 6 \cdot 6 = 36\$
Calculate the remaining expressions!
Exercise 7: Calculate!
a. \$5^3 =\$
b. \$4^4 =\$ (The fourth power of 4 is ...)
c. \$3^5 =\$ (The fifth power of 3 is ...)
d. \$2^6 =\$
e. \$12^2 =\$
f. \$20^2 =\$
g. \$15^2 =\$
h. \$(12 + 2)^2 =\$
j. \$(15 - 14)^{500} =\$
i. \$(20 - 7)^2 =\$
Observation:
As you can see, the exponent (the number on the upper right corner) tell us how much times we have to multiply the base (the big number).
Exercise 8: Calculate!
a. \$1^{500}\$
b. \$1^{280}\$
c. \$10^1 =\$
d. \$10^2 =\$
e. \$10^3 =\$
f. \$10^8 =\$
g. \$0^50 =\$
h. \$0^20 =\$
Observation:
As you can see from the exercise 8, we have find out that:
\$1^n = 1\$ (Remember: n is a variable. Variables represents all the numbers.)
\$0^n = 0\$
\$10^n = 1000...000\$ ( \$n\$ numbers)
Exercise 9: Write the following numbers as a power (there are several choices).
a. \$1000000 =\$
b. \$25 =\$
c. \$125 =\$
d. \$16 = \$
Exercise 10: Make a table with the squares of numbers from 1 to 10. I suggest you to learn it, because it's useful in plenty of exercises.
Exercise 11: Express in power:
a. \$a \cdot a \cdot a =\$
b. \$b \cdot b \ cdot b ... \cdot b =\$ (15 times)
c. \$2a \cdot 2a \cdot 2a =\$
d. \${(l^2)}^2 =\$
Exercise 12: Calculate the powers, like the example:
a. \$(2a)^2 = 2^2a^2 = 4a^2\$
b. \$(4b)^3 =\$
c. \$(2ab)^2 =\$
d. \${(ab)^2}^3\$
e. \$(2a + 3a)^4\$
Exercise 13: Calculate the following expressions:
a. \$(2a + 2b) + (3a + 3b) =\$
b. \$(5b + 4c) + (3a + 12) + (3c + 5 + a) + 2b\$
Remember: \$a + (b) = a + b\$
c. \$(2a + 3b) + (2a - 3b) =\$
d. \$(2a + 3b) - (2a + 3b) =\$
Remember: \$a - (b) = a - b\$
e. \$(2a + 3b) - (2a - 3b) =\$
Conclusion: As you can see, if a variable is on parenthesis, we following this rule:
If the operation before the parenthesis is plus, then we take out the parenthesis without any change.
If the operation before the parenthesis is minus, then we take out the parenthesis and we change the sign of the variables.
