$$
\begin{array}{|c|c|c|}
\hline
\textbf{Rule} & \textbf{Arithmetic example} & \textbf{Algebraic example} \\ \hline
(a^m) (a^n) = a^{m+n} & (5^{3})(5^5) = 5^{3+5}=5^8 & (x^6)(x^{-4})=x^{6+(-4)} = x^2 \\ \hline
(a^m)^n = a^{mn} = (a^n)^m &(2^2)^3=2^{6}=64 &(3z^2)^3=(3^3)(z^2)^3=27z^{6} \\ \hline
a^{-n} = \dfrac{1}{a^n} \quad (a \neq 0)& 2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8} & x^{-3} = \dfrac{1}{x^3} \\ \hline
\dfrac{a^m}{a^n} = a^{m-n} & \dfrac{7^{8}}{7^5} = 7^{8-5}=7^3 & \dfrac{x^5}{x^{-4}}=x^{5-(-4)} = x^9\\ \hline
a^0=1 & (-5)^0=1 & x^0=1 \quad (x \neq 0) \\ \hline
(a \times b)^n = (a^n)(b^n) & (2\times 5)^6=(2^6)(5^6) =10^6 & (2x)^3 = (2)^3(x)^3= 8x^3\\ \hline
\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} & \left(\dfrac{3}{2}\right)^3 = \dfrac{3^3}{2^3}=\dfrac{27}{8} & \left(\dfrac{2x}{3y^{2}}\right)^3 = \dfrac{2^3 x^3}{3^{3} (y^{2})^3}=
\dfrac{8x^3}{27 y^{6}}\\ \hline
\end{array}
$$

Exponents: Algebraic Manipulation

You will be expected to rewrite and manipulate algebraic expressions containing exponent terms. Here I list some examples of common manipulations:
$$8^x = (2^3)^x = 2^{3x}$$ $$27^x = (3^3)^x = 3^{3x}$$ $$3^{6x} = (3^2)^{3x} = 9^{3x}$$ $$3^{6x} = (3^3)^{2x} = 27^{2x}$$ $$9(3^x) = (3^2)(3^x) = 3^{x+2} \quad \text{Note:} \quad 9(3^x) \neq 27^{x}$$ $$8(2^x) = (2^3)(2^x) = 2^{x+3} \quad \text{Note:} \quad 8(2^x) \neq 16^{x}$$ $$\dfrac{3^{x+1}}{3} =\dfrac{3^{x+1}}{3^1}= 3^{(x+1)-1} = 3^x \quad \text{Note:} \quad \dfrac{3^{x+1}}{3} \neq x+1$$ $$ \dfrac{9^x}{27^y} = \dfrac{(3^2)^x}{(3^3)^y} = \dfrac{3^{2x}}{3^{3y}} = 3^{2x-3y} $$

Exponents: Adding and Subtracting Terms

You will be asked to simplify exponent expressions such as $2^{8}+2^{8}$. The most common mistake students make is to add the exponents, $2^8 + 2^8 \neq 2^{16}$, instead $2^8 + 2^8 = 2^{9}$. There is no general exponent rule when adding powers of numbers that have the same base, however, there are cases where simplification is possible using other rules of arithmetic. If you see a question on the SAT that asks you to add exponent terms with the same bases, the best approach is to factor the largest common term which will almost always lead to simplification. For example:
$$2^8 + 2^8 = 2^8(1 + 1) = 2^8(2) = 2^8(2^1) = 2^{8+1} = 2^9$$

Additional Examples

$$ 2^4 + 2^4 + 2^4 + 2^4 = 2^4(1 + 1 + 1 + 1) = 2^4(4) = 2^4(2^2) = 2^{4+2} = 2^6 $$ $$ 2^{20} – 3(2^{18}) = 2^{18}(2^2 – 3) = 2^{18}(4-3) =2^{18} $$ $$ 27^{\frac{5}{3}} – 8(27) = 27(27^{\frac{5}{3} – 1} – 8) = 27(27^{\frac{2}{3}} – 8)
= 27[ (3^3)^{\frac{2}{3}} – 8] = 27( 3^2 – 8) = 27 $$ $$ 2^{x} + 2^{x+1} = 2^x + (2^{x})(2^1) = 2^{x}(1+2) =3(2^x)$$

Radicals and Fractional Exponents

• A square root of a number $n$ is a number that, when squared, is equal to $n$ or $(\sqrt{n})^2 = n$.
• An exponent of $\dfrac{1}{2}$ is the same as taking the square root of a number, $\sqrt{n} = n^{\frac{1}{2}}.$
• A cube root of a number $n$ is a number that, when cubed, is equal to $n$ or $(\sqrt[3]{n})^3 = n$.
• An exponent of $\dfrac{1}{3}$ is the same as taking the cube root of a number, $\sqrt[3]{n} = n^{\frac{1}{3}}$.

$$
\begin{array}{|c|c|}
\hline
\textbf{Rule} & \textbf{Example} \\ \hline
(\sqrt{a})^2 = a \quad (a>0) & (\sqrt{7})^2 = 7 \\ \hline
\sqrt{a^2} = a \quad (a>0) & \sqrt{7^2} = 7 \\ \hline
\sqrt{a \times b} = \sqrt{a} \times \sqrt{b} & \sqrt{6}\sqrt{8}=\sqrt{48} = \sqrt{(16)(3)} = \sqrt{16}\sqrt{3} = 4\sqrt{3} \\ \hline
\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} & \dfrac{\sqrt{6}}{\sqrt{24}} = \sqrt{\dfrac{6}{24}} = \sqrt{\dfrac{1}{4}} = \dfrac{1}{2} \\ \hline
a^{\frac{m}{n}}= (a^{\frac{1}{n}})^m = (a^m)^{\frac{1}{n}}= (\sqrt[n]{a})^m=\sqrt[n]{a^m} & 27^{\frac{2}{3}} =
(27^2)^{\frac{1}{3}}=(27^{\frac{1}{3}})^2 = (\sqrt[3]{27})^2= (3)^2=9\\ \hline
\end{array}
$$

Here is a list of common exponent manipulations that you can expect on the SAT math section:
$$ \sqrt[6]{27} = 27^{\frac{1}{6}} = (3^3)^{\frac{1}{6}} = 3^{\frac{3}{6}} = 3^{\frac{1}{2}} = \sqrt{3}$$ $$ 64^{-\frac{2}{3}} = (2^6)^{-\frac{2}{3}} = 2^{6 \times -\frac{2}{3}} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16} $$ $$a^{\frac{3}{4}} = (a^{\frac{1}{4}})^3 = (\sqrt[4]{a})^3 = (a^3)^{\frac{1}{4}} = \sqrt[4]{a^3} $$ $$(\sqrt{x^{3}})^4 = [(x^{3})^{\frac{1}{2}}]^4 = x^{(3)(\frac{1}{2})(4)} = x^6$$ $$x \sqrt{x}=(x^1)(x^{\frac{1}{2}}) = x^{1+\frac{1}{2}} = x^{\frac{3}{2}}$$ $$ \sqrt[3]{8x^6} = (8x^6)^{\frac{1}{3}} = (8)^{\frac{1}{3}} (x^6)^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} (x)^{\frac{6}{3}} = 2x^2 $$ $$ \sqrt[6]{x^4 y^3 z^2} =(x^4 y^3 z^2)^{\frac{1}{6}} = x^{\frac{4}{6}} y^{\frac{3}{6}} z^{\frac{2}{6}} = x^{\frac{2}{3}} y^{\frac{1}{2}} z^{\frac{1}{3}} $$

Common Mistakes

Finally, I list the most common mistakes that students make when they have to simplify algebraic expressions with exponents.
$$
\begin{array}{|c|c|}
\hline
\textbf{Mistake} & \textbf{Example} \\ \hline
(3a)^3 \neq 3a^3 & \text{Instead} \quad (3a)^3 = (3^3)(a^3) = 27a^3 \\ \hline
\sqrt{a^2 + b^2} \neq a + b & \sqrt{3^2 + 4^2} \neq 3 + 4 \quad \text{instead} \quad \sqrt{3^2+4^2}=\sqrt{25}=5 \\ \hline
(a + b)^n \neq a^n + b^n & (2 + 3)^2 = 5^2 = 25 \neq 2^2 + 3^2 = 13 \\ \hline
(-a)^2 \neq -(a^2) & (-3)^2 = 9 \neq -(3^2) = -9 \quad \text{instead} \quad (-a)^2 = a^2 \\ \hline
\end{array}
$$

Here I list several strategies that can help you curb careless mistakes during the SAT test:

• Read Carefully: Read the question very carefully and read it several times. On the difficult problems, you will not grasp the entire question on one reading. You may have to read it two or three times, or more. In general, harder questions require several readings.
• Stay organized: Do all of your scratch work in a systematic manner. Write in the blank area in the test booklet.
• Write legibly: Your work should be clear enough that you can read your own handwriting. This is helpful in situations when you end up with an answer that is not in one of the answer choices. This often happens when one makes a careless mistake. To spot your mistake it helps if your work is written in a clear and legible manner.
• Don’t use the Calculator: I know a lot of students are completely reliant on the calculator, and many people would disagree with me when I suggest not using the calculator. All of the SAT math questions are written in a way that they can be solved without the use of calculator, and on many questions it might be to your advantage not to use the calculator. The problem with doing your work on the calculator is that you cannot go back to check your steps if you made a mistake. In contrast, it is a lot easier to spot a mistake if you have the steps written in your test booklet.
• Redraw diagrams: On the SAT one does not need to redraw things, but I find redrawing helps me digest the problem and also help me see the solution.
• Slow Down: Don’t rush off to attack the problem immediately and don’t change the problem to what you think it is asking, be careful about that temptation.
• Recognize the Difficulty Level of a Question: Look at the Official SAT tests and recognize where the difficult questions are, generally at the end of each subsection. Keep an eye on the medium level questions where you are likely to trip on misreading the question. The easy/medium questions rely more on how the question is phrased, whereas the harder questions test advanced concepts and one is less likely to trip on verbiage.
• Reread the question at the end: Once you have completed the problem, reread the question to make sure you are answering what the question is asking for. For example, if you defined a variable $x$ to solve the problem, check to make sure the question is not asking for the value of $x-2$.