Radicals
Decide whether each statement is true or false and then prove that it's true, or disprove with a counter example.
Example 1: \(x^2+y^2=\left(x+y\right)^2\)
This statement is false. A counterexample is \(3^2+4^2\neq\left(3+4\right)^2\).
Note: The reason you have to prove a true statement, rather than just give an example that works, is because sometimes an example works for an equation, but not all values substituted into the variables would make the statement true. An example of this is substituting \(0\) and \(1\) in for \(x\) and \(y\) in the previous example: \(0^2+1^2=\left(0+1\right)^2\).
Example 2: \(x+x=2x\)
This statement is true.
Proof:
\(\begin{align}x+x&=2x\ & \ &\text{1) Given}\\
x(1+1)&=2x\ & \ &\text{2) Factor out an x}\\
x(2)&=2x\ & \ &\text{3) Simplify}\\
2x&=2x\ & \ &\text{4) Commutative Property of Multiplication}\end{align}\)
Your Turn!
Example 3: \(\sqrt{x}\cdot\sqrt{y}=\sqrt{xy}\)
Example 4: \(\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}\)
Example 5: \(\sqrt{x}+\sqrt{y}=\sqrt{x+y}\)
Example 6: \(\sqrt{x}-\sqrt{y}=\sqrt{x-y}\)
Example 7: \(\sqrt{x}+\sqrt{x}=\sqrt{x^2}\)
Example 8: \(\sqrt{x}+\sqrt{y}=2\sqrt{x}\)
Example 9: \(\sqrt{x\cdot y\cdot x}=x\sqrt{y}\)
Number Systems
Here's a recap of the different types of number systems you've learned so far.
Natural Numbers: These are counting numbers: \(1, 2, 3, 4, 5, 6...\)
Whole Numbers: This set of numbers includes \(0\) and the natural numbers. \(0, 1, 2, 3, 4, 5, 6...\)
Integers: These include \(0\), the natural numbers, their opposites. \(...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6...\)
Rational Numbers: Any number than can be written as a fraction.
Irrational Numbers: Any number than cannot be written as a fraction.
Real Numbers: The set of numbers containing both rational and irrational numbers.
Example 10: Decide which number system(s) apply to the given scenario.
A) The number of volleyballs in the gym during volleyball practice.
B) The temperature in Chicago over the course of a year.
C) The number of fingers and toes that you have.
D) The amount of money you have in the bank.
E) The score that the Blackhawks have during a game.
F) The circumference of a dog bowl.
G) \({\sqrt{1},\ \sqrt{2},\ \sqrt{3},\ \sqrt{4},\ \sqrt{5},...}\)
H) \({\sqrt{-4},\ \sqrt{-3},\ \sqrt{-2},\ \sqrt{-1},\ \sqrt{0},\ \sqrt{1},\ \sqrt{2},\ \sqrt{3},\ \sqrt{4},\ \sqrt{5},...}\)
Decide whether each statement is true or false and then prove that it's true, or disprove with a counter example.
Example 1: \(x^2+y^2=\left(x+y\right)^2\)
This statement is false. A counterexample is \(3^2+4^2\neq\left(3+4\right)^2\).
Note: The reason you have to prove a true statement, rather than just give an example that works, is because sometimes an example works for an equation, but not all values substituted into the variables would make the statement true. An example of this is substituting \(0\) and \(1\) in for \(x\) and \(y\) in the previous example: \(0^2+1^2=\left(0+1\right)^2\).
Example 2: \(x+x=2x\)
This statement is true.
Proof:
\(\begin{align}x+x&=2x\ & \ &\text{1) Given}\\
x(1+1)&=2x\ & \ &\text{2) Factor out an x}\\
x(2)&=2x\ & \ &\text{3) Simplify}\\
2x&=2x\ & \ &\text{4) Commutative Property of Multiplication}\end{align}\)
Your Turn!
Example 3: \(\sqrt{x}\cdot\sqrt{y}=\sqrt{xy}\)
Example 4: \(\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}\)
Example 5: \(\sqrt{x}+\sqrt{y}=\sqrt{x+y}\)
Example 6: \(\sqrt{x}-\sqrt{y}=\sqrt{x-y}\)
Example 7: \(\sqrt{x}+\sqrt{x}=\sqrt{x^2}\)
Example 8: \(\sqrt{x}+\sqrt{y}=2\sqrt{x}\)
Example 9: \(\sqrt{x\cdot y\cdot x}=x\sqrt{y}\)
Number Systems
Here's a recap of the different types of number systems you've learned so far.
Natural Numbers: These are counting numbers: \(1, 2, 3, 4, 5, 6...\)
Whole Numbers: This set of numbers includes \(0\) and the natural numbers. \(0, 1, 2, 3, 4, 5, 6...\)
Integers: These include \(0\), the natural numbers, their opposites. \(...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6...\)
Rational Numbers: Any number than can be written as a fraction.
Irrational Numbers: Any number than cannot be written as a fraction.
Real Numbers: The set of numbers containing both rational and irrational numbers.
Example 10: Decide which number system(s) apply to the given scenario.
A) The number of volleyballs in the gym during volleyball practice.
B) The temperature in Chicago over the course of a year.
C) The number of fingers and toes that you have.
D) The amount of money you have in the bank.
E) The score that the Blackhawks have during a game.
F) The circumference of a dog bowl.
G) \({\sqrt{1},\ \sqrt{2},\ \sqrt{3},\ \sqrt{4},\ \sqrt{5},...}\)
H) \({\sqrt{-4},\ \sqrt{-3},\ \sqrt{-2},\ \sqrt{-1},\ \sqrt{0},\ \sqrt{1},\ \sqrt{2},\ \sqrt{3},\ \sqrt{4},\ \sqrt{5},...}\)