New exponents course

content/courses/.stackedit-trash/4.md

@@ -0,0 +1,27 @@
+---
+type: problem
+answers:
+  - $\frac{p^tq^t}{r^ts^t}$
+
+---
+
+@statement
+Expand and simplify $\left(\frac{x}{\frac yz}\right)^x$. 
+
+@hint
+Rewrite $\frac{pq}{rs}$ as the product of two fractions.
+
+@hint
+Use both the Power of a Product Rule and the Power of a Quotient Rule.
+
+@solution
+We write $\frac{pq}{rs}$ as $\frac pr \cdot \frac qs$. Then, we apply the Power of a Product and Power of a Quotient Rules.
+$$\left(\frac{pq}{rs}\right)^t
+= \left(\frac pr \cdot \frac qs\right)^t
+= \left(\frac pr\right)^t\left(\frac qs\right)^t
+= \left(\frac{p^t}{r^t}\right)\left(\frac{q^t}{s^t}\right)
+=\frac{p^tq^t}{r^ts^t}$$
+Note that there are many different ways to solve the problem, and they all lead to the same solution.
+<!--stackedit_data:
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+-->

content/courses/calculus/differential/introduction/index.md

@@ -133,8 +133,13 @@ And, now that $\delta x$ is out of the denominator, we can substitute
 $\delta x =0$:
 
 $$\frac{\delta y}{\delta x}=2x$$
+title: Introduction
+description: An intuitive introduction to derivatives
+index: 6
 
+---
+Okay, so now you're a whiz at limits, but you still don't feel like you're doing Calculus!
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content/courses/exponents/index.md

@@ -7,9 +7,22 @@ author: Sophia Zhu
 thumbnail: "/img/exponents-and-more.jpg"
 
 ---
-Exponents and logarithms are *power*-ful tools in mathematics. You should have a *base*-ic understanding of them if you want to learn topics like calculus.
+Exponents and logarithms are *power*-ful tools in mathematics. You must have a *base*-ic understanding of them if you want to learn topics like calculus.
 
-Alright, enough puns. On with the course!
+Alright, enough puns. Before you begin this course, brush up on the following topics if you're not very familiar with them:
+
+- Arithmetic with multi-digit numbers, negative numbers, fractions, and decimals. 
+- Properties of addition, subtraction, multiplication, and division
+- Variables
+- Simplifying expressions such as fractions
+
+This course covers the following topics:
+- properties, graphs, and real-world applications of exponents and logarithms
+- scientific notation
+
+These topics encompass a wide range of difficulties. If the course is too easy, feel free to skip lessons. And if it's too difficult, try practicing some other math topics before coming back to this course. 
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content/courses/exponents/exponents-basic/introduction/index.md → content/courses/exponents/properties/introduction/index.md

@@ -10,14 +10,18 @@ Sometimes, we want to multiply a number by itself many times. Instead of having
 $$3^5=3\cdot3\cdot3\cdot3\cdot3$$
 (read as 3 to the power of 5, 3 to the 5th power, or 3 to the 5th)
 
-In the example above, $3^5$ means "multiply 3 by itself 5 times." The "$5$" is called the *exponent,* and the "$3$" is called the *base.* A base together with its exponent is called a *power*.
+In the example above, $3^5$ means "multiply 3 by itself 5 times." The small "$5$" in the top right is called the *exponent,* and the "$3$" in the bottom left is called the *base.* A base together with its exponent is called a *power*.
 
-To type a power on a computer, type {base}^{exponent}. For example, for $3$^$5$ we would type $3$^$5$ or {$3$}^{$5$}. 
+To type a power on a computer, type {base}^{exponent}. For example, for $3$^$5$ we would type $3$^$5$. On YMath, you might also need to use your keyboard's left and right arrow keys to move your cursor in or out of an exponent. 
 
 Let's look at another example. 
 $$(x-1)^{2y}$$
 The base is $x-1$, and the exponent is $2y$. So, this expression means to multiply $x-1$ by itself $2y$ times.
 
+@1
+
+@2
+
 Now for an interesting question: what is $3^0$? Well, we are multiplying $3$ by itself $0$ times, so the answer is $0$, right? Not quite--we'll use something called an identity element to show why that isn't the case. 
 
 ### Identity Element
@@ -45,17 +49,19 @@ Notice that $3^0$ actually equals $1$! In the following lessons, you will learn
 The *Order of Operations* is a standard order of evaluating an expression. Many students use the mnemonic PEMDAS to remember that expressions in **p**arentheses are evaluated first, followed by **e**xponents, **m**ultiplication and **d**ivision, and **a**dition and **s**ubtraction. However, be very careful if you use PEMDAS. Multiplication and division are considered to be on the same level, meaning that both $3\cdot4\div5$ and $3\div4\cdot5$ are evaluated from left to right. Similarly, addition and subtraction are on the same level, so $3+4-5$ and $3-4+5$ are evaluated from left to right.
 
 For "towers" or "stacks" of exponents, evaluate the expression from top to bottom. For instance, 
-$$2^{3^0}=2^{(3^0)}=2^1=2$$
-$$2^{3^0}\neq(2^3)^0=8^0=1$$
-$$w^{x^{y^z}}=w^{(x^{(y^z)})}$$
+$$2^{3^0}=2^{\left(3^0\right)}=2^1=2$$
+$$2^{3^0}\neq\left(2^3\right)^0=8^0=1$$
+$$w^{x^{y^z}}=w^{\left(x^{\left(y^z\right)}\right)}$$
 
 ### Warnings
 - Usually, $x^y$ is not equal to $y^x$. For example, $2^3=8$ and $3^2=9$. This is different from addition and multiplication, where you can switch the numbers around. 
 - Be careful with the Order of Operations. Exponents come before all operations except parentheses!
-$$x\cdot y^z=x\cdot(y^z)​\neq (x\cdot y)^z$$
-$$-y^z=-1\cdot(y^z)​\neq (-y)^z$$
-The second example shows a very common error. A negative sign is the same as multiplying by $-1$, so make sure you evaluate the exponent before the negative sign.
+$$x\cdot y^z=x\cdot\left(y^z\right)​\neq \left(x\cdot y\right)^z$$
+
+@3
+
+Now that we know how to evaluate exponents (and how not to evaluate them), we are ready to practice our new skills!
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content/courses/exponents/properties/introduction/problems/1.md

@@ -0,0 +1,22 @@
+---
+type: problem
+answers:
+  - "8"
+
+---
+
+@statement
+Evaluate $2^3$.
+
+@hint
+Which number is the base? Which is the exponent?
+
+@hint
+Try rewriting the expression as repeated multiplication. 
+
+@solution
+The bottom-left number ($2$) is the base and the top-right number ($3$) is the exponent. In a power, the exponent tells you how many times to multiply the base by itself. So, we multiply $2$ by itself $3$ times.
+$$2^3=2\cdot2\cdot2=8$$
+<!--stackedit_data:
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content/courses/exponents/properties/introduction/problems/2.md

@@ -0,0 +1,20 @@
+---
+type: problem
+answers:
+  - "11^5"
+
+---
+
+@statement
+Write $11\cdot11\cdot11\cdot11\cdot11$ in exponential form.
+
+@hint
+What's the base? What's the exponent?
+
+@solution
+We see that $11$ is multiplied by itself $5$ times, so $11$ is the base, and $5$ is the exponent. In exponential form, we write this as $11^5$, and we type it as 11^5.
+
+
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content/courses/exponents/properties/introduction/problems/3.md

@@ -0,0 +1,26 @@
+---
+type: problem
+answers:
+  - "-1"
+
+---
+
+@statement
+Evaluate $-1^4$.
+
+@hint
+Is the negative sign part of the power?
+
+@hint
+Remember the Order of Operations -- multiplication comes after exponentiation.
+
+@solution
+The expression $-1^4$ is equivalent to $-1\cdot1^4$. By the Order of Operations, we find that 
+$$-1\cdot1^4=-1\cdot(1\cdot1\cdot1\cdot1)
+=-1\cdot1=-1$$
+Note that $-1^4\neq(-1)^4=1$.
+
+
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content/courses/exponents/properties/power-to-a-power/index.md

@@ -0,0 +1,45 @@
+---
+type: lesson
+index: 5
+title: Power to a Power Property
+
+---
+
+This is the last property we'll be learning in this chapter. Get ready for lots of powers with the Power to a Power property!
+
+- Power to a Power: $(x^a)^b=x^{ab}$
+
+
+## Power to a Power
+Is there a way to simplify $(x^a)^b$? Let's first substitute some values so that we can write the powers as repeated multiplication.
+
+$$(x^2)^3
+=(x^2)\cdot(x^2)\cdot(x^2)
+=(x\cdot x)\cdot(x\cdot x)\cdot(x\cdot x)
+=x^6$$
+
+There are $3$ groups of $2$ $``x"$s being multiplied together, which makes $2\cdot3=6$ $``x"$s in total. 
+
+Now we can simplify the general expression.
+
+$$(x^a)^b=\underbrace{(x^a)(x^a)\dots(x^a)}
+_\text{$b$ times}$$
+$$= \underbrace{
+	\underbrace{(xxx \ldots x)}_\text{$a$ times}
+	\underbrace{(xxx \ldots x)}_\text{$a$ times} \ldots 
+	\underbrace{(xxx \ldots x)}_\text{$a$ times}
+}_\text{$b$ times}$$
+$$=\underbrace{xxx \ldots x}_\text{$ab$ times}
+=x^{ab}$$
+
+In $(x^a)^b$, there are $b$ groups of $``\left(x^a\right)"$s being multiplied together. This makes a total of $ab$ $``x"$s multiplied together. Therefore, $(x^a)^b=x^{ab}$.
+
+## Warnings
+Recall from the first lesson that towers of exponents are evaluated from the top down. So, in general,
+$$x^{y^z}=x^{\left(y^z\right)}\neq \left(x^y\right)^z=x^{yz}$$
+$$x^{y^z}\neq x^{yz}$$
+
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content/courses/exponents/properties/review/index.md

@@ -0,0 +1,127 @@
+---
+type: lesson
+index: 7
+title: Properties
+
+---
+
+You might remember that when you learned about multiplication, you learned about some properties like the Associative Property, Commutative Property, and the Distributive Property. Now, we'll be looking at some properties of exponents.
+
+### List of Exponent Properties
+- Product of Powers: $x^a\cdot x^b=x^{a+b}$
+- Quotient of Powers: $\frac{x^a}{x^b}=x^{a-b}$
+- Power to a Power: $(x^a)^b=x^{ab}$
+- Power of a Product: $(xy)^a=x^a\cdot y^a$
+- Power of a Quotient: $\left(\frac{x}{y}\right)^a=\frac{x^a}{y^a}$
+
+
+## Product of Powers
+Let's start with a simple question. What is 
+$2^5$?
+
+We know a way to evaluate powers -- write them as multiplication! 
+
+$$2^5=2\cdot2\cdot2\cdot2\cdot2=32$$
+
+So, $2^5=32$. How about another question: what is $2^2\cdot2^3$?
+
+We can solve this using the Order of Operations. We evaluate
+$$2^2\cdot2^3=(2\cdot2)(2\cdot2\cdot2)=(4)(8)=32$$
+
+Notice anything? It's the same answer as before! Is it a coincidence?
+
+It turns out that a very important clue is in the first step of evaluating both expressions. Take a look:
+$$2\cdot2\cdot2\cdot2\cdot2$$
+$$(2\cdot2)(2\cdot2\cdot2)$$
+
+These expressions look very similar! In fact, they're equal by the Associative Property of Multiplication. This property states that if your expression only has multiplication in it, it doesn't matter where you put the parentheses. The product will always be the same. 
+
+Now we know why $2^2\cdot2^3=2^5$. Let's generalize this a bit more. Is there a way we can rewrite $x^2\cdot x^3$? Let's try:
+
+$$x^2\cdot x^3
+=(xx)(xxx)
+=xxxxx
+=x^5$$
+
+Nice! We just showed that $x^2\cdot x^3 = x^5$. This is because $x^2$ is the product of $2$ $``x"$s, and $x^3$ is the product of $3$ $``x"$s, so the product of $x^2$ and $x^3$ is the product of $5$ $``x"$s in total. 
+
+Let's make an even more general statement! What is $x^a \cdot x^b$? Even though we don't know the values of $a$ and $b$, we can still write out the powers.
+
+$$x^a \cdot x^b = 
+	(\underbrace{xxx\ldots x}_\text{$a$ times})
+	(\underbrace{xxx\ldots x}_\text{$b$ times})$$
+$$= (\underbrace{xxx\ldots x}_\text{$a+b$ times})= x^{a+b}$$
+
+Since $x^a$ is the product of $a$ $``x"$s, and $x^b$ is the product of $b$ $``x"$s, the product of $x^a$ and $x^b$ is the product of $(a+b)$ $``x"$s.
+
+Awesome, we've just proved the Product of Powers Property! (Actually, we've shown that it works for integers, which is good enough for now.) Can you see why it's called the Product of Powers Property?
+
+### Warnings
+The bases of the powers have to be the same for this property to work. If you try to simplify $x^2\cdot y^3$, you will not get anywhere since it's already simplified.
+$$x^2 \cdot y^3=(xx)(yyy)$$
+$$\neq x^5\neq y^5 \neq (x+y)^5 \neq (xy)^5$$
+In case you missed it, all the expressions on the second line are **not equal** to the expressions on the first line.  Don't make up new properties unless you know that they're true. Chances are, the property will be false! 
+
+Speaking of false properties, make sure the powers are being multiplied, not added together. 
+$$x^2+x^3\neq x^5$$
+
+If you're curious, a valid way to rewrite the above expression is to use the Product of Powers Property along with the Distributive Property.  
+$$x^2+x^3=x^2\cdot1+x^2\cdot x^1=x^2(1+x)$$
+
+## Quotient of Powers
+The Quotient of Powers Property is very similar to the Product of Powers Property, except, as you probably guessed, it involves quotients instead of products. It lets us easily simplify expressions like $\frac{x^5}{x^2}$.
+
+Speaking of which, what is $\frac{x^5}{x^2}$? Let's write it out. 
+$$\frac{x^5}{x^2}
+=\frac{\cancel {xx}\cdot xxx}{\cancel {xx}}
+=x^3$$
+
+Let's generalize this a bit more. How can we rewrite $\frac{x^a}{x^b}$, where $a\geq b$?
+
+$$\frac{x^a}{x^b}
+= \frac{\overbrace{xxx \ldots x}^\text{$a$ times}}
+{\underbrace{xxx \ldots x}_\text{$b$ times}}$$
+
+$$= \frac
+	{\overbrace{\cancel{xxx \ldots x}}^\text{$b$ times}\cdot 
+	\overbrace{xxx \ldots x}^\text{$a-b$ times}}
+	{\underbrace{\cancel{xxx \ldots x}}_\text{$b$ times}}$$
+
+$$=x^{a-b}$$
+
+Since $a\geq b$, there are at least as many $``x"$s in the numerator as the denominator, so all the $``x"$s in the denominator will cancel out. So, there will be $(a-b)$ $``x"$s left over, which equals $x^{a-b}$.
+
+We just showed that $\frac{x^a}{x^b}=x^{a-b}$!
+
+## Power to a Power
+Is there a way to simplify $(x^a)^b$? Let's first substitute some values so that we can write the powers as repeated multiplication.
+
+$$(x^2)^3
+=(x^2)\cdot(x^2)\cdot(x^2)
+=(x\cdot x)\cdot(x\cdot x)\cdot(x\cdot x)
+=x^6$$
+
+There are $3$ groups of $2$ $``x"$s being multiplied together, which makes $2\cdot3=6$ $``x"$s in total. 
+
+Now we can simplify the general expression.
+
+$$(x^a)^b=\underbrace{(x^a)(x^a)\dots(x^a)}
+_\text{$b$ times}$$
+$$= \underbrace{
+	\underbrace{(xxx \ldots x)}_\text{$a$ times}
+	\underbrace{(xxx \ldots x)}_\text{$a$ times} \ldots 
+	\underbrace{(xxx \ldots x)}_\text{$a$ times}
+}_\text{$b$ times}$$
+$$=\underbrace{xxx \ldots x}_\text{$ab$ times}
+=x^{ab}$$
+
+In $(x^a)^b$, there are $b$ groups of $``\left(x^a\right)"$s being multiplied together. This makes a total of $ab$ $``x"$s multiplied together. Therefore, $(x^a)^b=x^{ab}$.
+
+## Power of a Product
+Coming soon!
+
+## Power of a Quotient
+Coming soon!
+<!--stackedit_data:
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