Exponents

On September 30, 1868, Louisa May Alcott published the first volume of Little Women.  The story was extremely successful and has been beloved by readers ever since.  Louisa May Alcott would make an excellent historical figure to use as an example on your SAT writing section.  You can read more about her life here, and if you have read Little Women, remember that it could make an excellent literary example too!

9/30 Exponents

Read the following SAT test question and then select the correct answer.

Math questions need to be read just as carefully as reading questions. Avoid incomplete answers by making a note of the bottom line. Are you solving for x, or do you need the answer to 2x + 3? Assess your options for solving the problem, choose the most efficient method, and attack the problem! Once you have the answer, loop back to verify that it addresses the bottom line.

If , which of the following expresses a in terms of b?

Bottom line: This question asks you to solve for the variable a.

Assess your options: You could try to plug in the answer choices for a and choose a number for b to try to find the answer. However, that method requires you to work several problems and includes multiple steps. Instead, use algebra to rearrange the equation.

Attack the problem: You see two numbers with exponents. When two bases are the same, then the exponents can be set equal to each other. Your two bases are 2 and 4. How can you make both bases the same? Use the fact that 2² = 4 by plugging that into your equation.

$2^{a} = 4^{b}$

$2^{a} = 2^{2b}$

Now you can ignore the bases and set the exponents equal to each other. You now have:

a = 2b

Loop back: The question asked you to find a in terms of b, and that is just what you did. Look down at your answer choices

(A)

(B) b

(C) 2b

(D) 4b
(E)

On sat.collegeboard.org, 54% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Equations

Read the following SAT test question and then select the correct answer.

Read the question carefully, identify the bottom line (what the question is asking), and assess your options for solving it. You want to be as efficient as possible when solving math questions, so for most problems you should not look at the multiple choice answers before attacking the problem with the method you have chosen. Always loop back at the end of the problem to make sure that your answer addresses the bottom line.

If $(t-2)^{2}=0$, what is the value of (t + 3)(t + 6)?

You must find the value of (t - 3)(t + 6). In order to do this, you must first find the value of t. Paraphrase the question in your mind: “If this is true, then solve this.” This question is already set up in two steps for you.  Solve the first equation and you will have the key to solving the second part of the problem because there is only one variable involved: t.

Think about the first equation logically. Something squared is equal to zero, so what can be multiplied by itself and equal zero? The only possible answer is zero! The squared portion of the problem must be equal to zero.

$(t-2)^{2}=0$ and $0^{2}=0$, therefore t - 2 = 0.

When you add the 2 to both sides of your new equation, you will see that = 2. Now that you know the value of t, you have all the information that you need to solve the second part of the problem with simple arithmetic.

(t + 3)(t + 6)

(2 + 3)(2 + 6)

(5)(8)

40

Loop back to make sure that the answer you found answers the question you were asked. The problem asked for the value of (t + 3)(t + 6), and that is exactly what you found. Finally, match your answer to the correct answer choice.

(A) 40

(B) 18

(C) 9

(D) 4

(E) It cannot be determined from the information given.

On sat.collegeboard.org, 58% of responses were correct.

For more help with math, visit www.myknowsys.com!

Exponents

Pursuing your dreams takes dedication, particularly when you have to juggle athletic training and school at the same time. 16-year-old Olympic athlete Ariel Hsing has been dreaming of playing table tennis in the Olympics since she was just 8 years old. Her talent and dedication to her sport have helped to make that dream a reality. However, she understands that table tennis is not as popular as other sports (and therefore the endorsements are less lucrative). In addition to striving for Olympic greatness, Hsing also aspires to attend Stanford. You can read more about Ariel Hsing here. Her dedication to both her sport and her academics make her a great "Excellent Example" for your essay.

7/17 Exponents

Read the following SAT test question and then select the correct answer.

As always, remember to follow the Knowsys method for math. Read the problem carefully and identify the bottom line (what you are looking for). Then, consider your options. How could you solve it? How should you solve it? Next, attack the problem using the method that you selected. Finally, loop back and verify that your answer matches the bottom line.

If  and , and if , which of the following is true?

The key to this question is to know your exponent rules and follow the Knowsys method for math. Start by reading the problem carefully and identifying the bottom line. If you glance at the answers below, you can see that you are looking for the relationship between x and y. Now, consider your options. What could you do? What should you do? Since you only have two variables in the equation (x and y), all you need to do is simplify the equation. Next, attack the problem using the method you have selected. You want to simplify the equation and move x and y out of the exponents. In order to do that, you need matching bases.  Using your exponent rules you can rewrite the equation as follows

$3^{2x}=27^{2y}\Rightarrow 3^{2x}=(3^{3})^{2y}\Rightarrow 3^{2x}=3^{3*2y}\Rightarrow 3^{2x}=3^{6y}$

Since the bases are the same, you can just eliminate them and you are left with

$3^{2x}=3^{6y} \Rightarrow 2x=6y$

Now, do a little basic algebra to simplify and get

$2x=6y \Rightarrow x=3y$

The final step in the Knowsys method for math is to loop back and verify that your answer matches the bottom line. Since you have followed all the algebra rules, you know that your equation is true. Look at the answers below and choose the one that matches your prediction.

(A) x=y
(B) x=2y
(C) 2x=y
(D) x=3y
(E) 3x=y

The correct answer choice is (D).

On sat.collegeboard.org 51% of the responses were correct.

For more help with math, visit www.myknowsys.com.

Integers

Pythagoras, best known to high school students for his Pythagorean Theorem, actually discovered much more than that one formula. Even if you are not mathematically inclined, the beginning of this paper has some interesting notes on how the Pythagoreans--the followers of Pythagoras--lived.

3/28 Integers

Always attempt to solve the problem before looking at the answer choices. Read carefully, then identify the bottom line--what the question is actually asking--and mark it at the top of your scratch work. Assess you options by asking "What could I do?" to open your toolbox, then "What should I do?" to select the best way to solve the problem. Attack the problem fearlessly, then loop back to the bottom line to check whether what you found is the correct answer.

If p is an odd integer, which of the following is an even integer?

At the top of your scratch work, write even = ?

Next, ask "What could I do?" You could think through each answer choice abstractly, determining that if p is odd then... but that is difficult and gets confusing quickly. You could pick a number for p, then use that number to find a value for each answer choice. The smallest odd number is the best for this. Pick one. Since this question includes the phrase "which of the following," the answer is very likely to be D or E. Start at the bottom and work your way up.

E) $p^{2}-p$
If p = 1, then $1^{2}-1=0$. 0 is neither positive nor negative, but neutral; however, it is still even. This distinction confuses some students, so make sure you know it. Now loop back to the bottom line. $p^{2}-p=0$, so it is even, so it is the answer. On the SAT, you could continue on from this point or check the other answers.

D) $(p-2)^{2}$
$(1-2)^{2}=(-1)^{2}=1$ is odd.

C) $p^{2}-2$
$1^{2}-2=1-2=-1$ is odd.

B) $p^{2}$
$1^{2}=1$ is odd.

A) $p-2$
$1-2=-1$ is odd.

On sat.collegeboard.org, 62% of responses were correct.

For more help with math, visit www.myknowsys.com.

Exponents

Remember to follow the Knowsys Method: read carefully, identify the bottom line, assess your options, attack the problem, and loop back to ensure that you answered the question correctly.

If $x^{\frac{1}{3}}=y^{2}$, which of the following must be equivalent to x?

After reading, find the bottom line and note it at the top of your scratch work.

x=?

Next, assess your options. There are two courses of action apparent here: you could pick numbers and plug them into x and y, or you could apply the exponent rules to solve for x. Which would be faster and easier? The exponent rules.

$x^{\frac{1}{3}}=y^{2}$

What can you do here? Since you need to isolate x, pay attention to its exponent. Normally, a fraction in an exponent indicates that you need to take a root--in this case, a cube root--but since you cannot take the root of a variable, do the opposite. Cube both sides.

$x^{\frac{1}{3}^{3}}=y^{2^{3}}$

The rule for "power to a power" situations, when an exponent is itself the base of another exponent, is to simply multiply the powers together.

$x=y^{6}$

Loop back to your bottom line. You were looking for x, and you found that  $x=y^{6}$. Now look at the answer choices.

$A)y^{\frac{1}{6}}$

$B)y^{\frac{2}{3}}$

$C)y^{\frac{3}{2}}$

$D)y^{3}$

$E)y^{6}$

On sat.collegeboard.org, 50% of responses were correct.

Want more help with math? Visit myknowsys.com!

Exponents

Follow the Knowsys Method and remember to read the problem, identify the bottom line, assess your options, and attack the problem. Then loop back to check that you answered the right question. For the vast majority of problems, you do not need to look at the answer choices before this point.

What is the largest possible integer value of n for which $5^{n}$ divides into $50^{7}$?

The bottom line is easy to find here: n=?

Now assess your options. You could look at the answer choices and plug them in, calculate each product, and see whether $50^{7}$ can divide by it evenly. But there must be a faster way! This is an exponent problem, so think about your exponent rules. If you can get the bases to match, finding the appropriate value of n will be easy.

Fortunately, 50 is a multiple of 5. It is also a multiple of 25.

$50=2(5^{2})$

Therefore,

$50^{7}=(2(5^{2}))^{7}$

Now you can apply the distributive property and the exponent rules.

$50^{7}=2^{7}(5^{2^{7}})$

$50^{7}=2^{7}(5^{14})$

Now you know that $(5^{14})$ is a product of  $50^{7}$. There's not much you can do from here, so look at the answer choices.

(A) 2
(B) 7
(C) 9
(D) 10
(E) 14

On sat.collegeboard.org, 37% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

Exponents

Mathematics: Exponents

According to the Knowsys Method, the first thing you should do when facing a math problem is to read carefully, then note the bottom line, assess your options, and attack the problem.

If $x^{-2}=16$, what is the value of $x^{2}$?

At the top of your scratch work, write the bottom line.  $x^{2}=?$

In most problems you would want to solve for x, but here you need to solve for $x^{2}$ instead. How do you do that?

First, notice that the given exponent is very similar to what you need to find. To correct negative exponents, convert each term to its reciprocal. Since you do not have a fraction to start with, simply put 1 in the numerator.

$\frac{1}{x^{2}}=16$

To get the variable out of the denominator, multiply both sides by $x^{2}$

$x^{2}(\frac{1}{x^{2}})=(16)x^{2}$

After canceling out $x^{2}$ on the left-hand side of the equation, you are left with $1=16x^{2}$  Divide both sides by 16 to isolate $x^{2}$

$\frac{1}{16}=\frac{16x^{2}}{16}$

$\frac{1}{16}=x^{2}$

Now loop back to your bottom line and check whether you answered the correct question.

$x^{2}=?$

Look at the answer choices to make a final selection:

$A) \frac{1}{16}$

$B) \frac{1}{4}$

$C) 2$

$D) 4$

$E)12$