Solving Equations
1.1 Prerequisites and Review
Exponents Refresher
Before we solve bigger algebra problems, we need to revisit some notation that may have been forgotten. When repeatedly multiplying a number, we simplify this by using an exponent.
Definition: Base and Exponent
For a real number $x$ and a positive integer $n$, the expression $x^n$ means multiply $x$ by itself $n$ times.
$$x^n=\underbrace{x\cdot x\cdot x\cdots x}_{n\text{ factors}}.$$
$$2\cdot 2 \cdot 2\cdot 2\cdot 2\cdot 2 = 2^5$$
$2$, multiplied by itself five times is written $2^5$. When using this notation, several rules reveal themselves that can help with calculations. For example, if we see an expression that has two numbers with the same base, thats the larger number that isn't the exponent, multiplied together, we end up adding the exponents.
$$3^5 \cdot 3^2 =\underbrace{(3\cdot 3\cdot 3\cdot 3\cdot 3)}_{\text{five 3's}} \cdot \overbrace{(3\cdot 3)}^{\text{two more}}=3^7$$
MyOpenMath: Product Rule
When we have an exponent raised to another exponent, those end up getting multiplied.
MyOpenMath: Power to a Power
Similarly, if the two numbers are being divided the exponents end up being subtracted because the division cancels some of the repeated multiplication.
$$\frac{3^5}{3^2}=\frac{3\cdot 3 \cdot 3 \cdot 3 \cdot 3}{3\cdot 3}=\frac{3\cdot 3 \cdot 3 \cdot \cancel{3 \cdot 3}}{\cancel{3\cdot 3}}=3^3$$
MyOpenMath: Quotient Rule
Exponents have many rules which are just shortcuts for fully expanding the product and reducing. These can save time and make calculations much easier.
Fact: Exponent Rules You Will Use Constantly
For nonzero real numbers $a,b$ and integers $m,n$:
- $a^m\cdot a^n=a^{m+n}$
- $\dfrac{a^m}{a^n}=a^{m-n}$
- $(a^m)^n=a^{mn}$
- $(ab)^n=a^n b^n$
- $\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}$
- $a^0=1$
- $a^{-n}=\dfrac{1}{a^n}$
Example: Simplify an exponential expression
Simplify
$$\frac{(2x^3y^{-1})^2}{4xy^{-3}}.$$
Show Solution
First expand the numerator:
$$ (2x^3y^{-1})^2=2^2x^6y^{-2}=4x^6y^{-2}. $$
Now divide by $4xy^{-3}$:
$$ \frac{4x^6y^{-2}}{4xy^{-3}}=x^{6-1}y^{-2-(-3)}=x^5y^1=x^5y. $$
Final answer:
$$x^5y. $$
MyOpenMath: Exponents and Fractions
MyOpenMath: Negative Exponents
MyOpenMath: All Together
Principal Roots and Radicals
Roots are the inverse of powers, but we have to be precise about which root symbol returns. For example, $\sqrt[2]{16}$ is the square root of 16. It asks: what is the number that when multiplied by itself two times results in 16. Because both $4\cdot 4 =16$ and $(-4)(-4)=16$, we need to be specific. the principle root is the positive solution. So we'll always use the positive result.
The $\sqrt[3]{8}$ is asking what number when multiplied by itself $3$ times yields $8$. The $n$, the index, in $\sqrt[n]{a}$ is telling you how many multiplications happen, and the $a$ is the result of that multiplication. When the index is not specified, like $\sqrt{4}$, we assume it to be 2.
Definition: Principal n-th Root
For $a\ge 0$ and integer $n\ge 2$, the principal n-th root $\sqrt[n]{a}$ is the nonnegative real number whose $n$th power is $a$.
Even roots will not always have real results. For example, $\sqrt{-9}$ doesn't make sense the way we've defined it here. What number can you multiply by itself twice to get $9$? Well, $3\cdot 3 = 9$, but its not negative. And $(-3)(-3)=9$ is also positive. In the real numbers, there is no way to square a number and it be less than zero. Odd roots, however, don't have an issue with this. $(-2)(-2)(-2)=-8$, so its perfectly reasonable to say $\sqrt[3]{-8}=-2$.
Fact: Even-vs-odd root behavior
- If $n$ is even, $\sqrt[n]{a}$ is real only when $a\ge 0$.
- If $n$ is odd, $\sqrt[n]{a}$ is real for all real $a$.
- $\sqrt[n]{a^n}=|a|$ for even $n$, and $\sqrt[n]{a^n}=a$ for odd $n$.
Example: Evaluate radicals carefully
Evaluate each expression.
- $\sqrt{49}$
- $\sqrt[3]{-27}$
- $\sqrt{x^2}$
Show Solution
- $\sqrt{49}=7$ (principal square root is nonnegative).
- $\sqrt[3]{-27}=-3$ (odd index allows negative values).
- $\sqrt{x^2}=|x|$, not just $x$.
Its also true that these roots, sometimes called radicals, play nicely with multiplication and division just like exponents. What I mean by that is that if you square root a product, its the same as square-rooting each factor of the product and multiplying them together.
$$\sqrt{4\cdot 13}=\sqrt{4}\cdot \sqrt{13}=2\sqrt{13}$$
It works for division, too.
$$\sqrt{\frac{3}{25}}=\frac{\sqrt{3}}{\sqrt{25}}=\frac{\sqrt{3}}{5}$$
It does not work for addition or subtraction. This is a very common mistake.
$$\sqrt{9+25}\neq \sqrt{9}+\sqrt{25}$$
Instead, $$\sqrt{9+25}=\sqrt{34}$$
Fact: Rules for Reducing Radicals
As long as $a$ and $b$ are positive real numbers, the following are true.
- $\sqrt{ab}=\sqrt{a}\sqrt{b}$
- $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
- $\sqrt{-a}=i\sqrt{a}$ where $i=\sqrt{-1}$
That last rule deals with complex numbers which we'll talk more about later. Right now, the first two rules are what we really need to reduce radicals.
Example: Reduce a radical
Reduce $\sqrt{27}$
Show Solution
Notice that $27=9\cdot 3$. So we can replace $27$ in the expression and use the first rule above to simplify the expression.
$$\sqrt{27}=\sqrt{9\cdot 3}=\sqrt{9}\sqrt{3}=3\sqrt{3}$$
MyOpenMath: Reducing a Radical
Combining Radical Terms
Radical expressions behave like like terms only when the radical part matches exactly.
Definition: Like Radical Terms
Two radical terms are like radicals when they have the same index and the same radicand after simplification.
Example: Combine radicals
Simplify
$$3\sqrt{2}+5\sqrt{2}-\sqrt{8}. $$
Show Solution
Rewrite $\sqrt{8}$:
$$\sqrt{8}=\sqrt{4\cdot 2}=2\sqrt{2}. $$
Now combine like radicals:
$$3\sqrt{2}+5\sqrt{2}-2\sqrt{2}=6\sqrt{2}$$
Final answer:
$$6\sqrt{2}. $$
MyOpenMath: Combining radical expressions
Evaluating Expressions
The next review topic is evaluationg expressions, or put more simply, plugging stuff in. Put numbers in carefully, then simplify in the right order.
Definition: Evaluate an Expression
To evaluate an algebraic expression means to substitute a value for each variable and simplify.
Example: Evaluate with order of operations
Evaluate
$$3x^2-2x+5$$
when $x=-2$.
Show Solution
Substitute $x=-2$:
$$3(-2)^2-2(-2)+5.$$
Compute step by step:
$$3(4)+4+5=12+4+5=21.$$
So the value is
$$21. $$
Example: Evaluate a multivariable expression
Evaluate
$$2a^2b-3ab^2$$
at $a=-1$ and $b=2$.
Show Solution
Substitute:
$$2(-1)^2(2)-3(-1)(2)^2=2(1)(2)-3(-1)(4).$$
Then simplify:
$$4+12=16.$$
So the value is
$$16. $$