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SAT Math : Pattern Behaviors in Exponents

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Example Question #268 : Exponents

Solve for .

$\ln x=\ln (x-1)-\ln (x+3)$

Possible Answers:

$no\ solution$

Correct answer:

Explanation:

If we combine into a single logarithmic function we get:

$\ln x=\ln \frac{x-1}{x+3}$

$x = \frac{x-1}{x+3}$

Solving for we get x=-1 .

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Example Question #269 : Exponents

If is the complex number such that i^2=-1 , evaluate the following expression:

$i^{25}-i^{23}+i^{21}-i^{19}+i^{17}-i^{15}.\ .\ .+i$

Possible Answers:

13i

-25i

12i

Correct answer:

13i

Explanation:

The powers of i form a sequence that repeats every four terms.

i¹= i

i² = -1

i³ = -i

i⁴ = 1

i⁵ = i

Thus:

i²⁵ = i

i²³ = -i

i²¹ = i

i¹⁹= -i

Now we can evalulate the expression.

i²⁵ - i²³+ i²¹- i¹⁹ + i¹⁷..... + i

= i + (-1)(-i) + i + (-1)(i) ..... + i

= i + i + i + i + ..... + i

Each term reduces to +i. Since there are 13 terms in the expression, the final result is 13i.

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Example Question #12 : Exponents

If $x^7y^8z^{10} < 0$ , then which of the following must also be true?

Possible Answers:

y < 0

zx > 0

x < 0

yz < 0

zy > 0

Correct answer:

x < 0

Explanation:

We know that the expression must be negative. Therefore one or all of the terms x⁷, y⁸ and z¹⁰ must be negative; however, even powers always produce positive numbers, so y⁸ and z¹⁰ will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x⁷ must be negative, so x must be negative. Thus, the answer is x < 0.

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Example Question #1 : How To Find Patterns In Exponents

Simplify the following:

$\frac{48^5^0+80^3^0}{4^2^0}$

Possible Answers:

17^5^0

2^1^6^0*3^5^0+2^8^0*5^3^0

4^1^6*3^2^5+2^2^0*5^1^5

12^3^0+5^1^0

17^8^0

Correct answer:

2^1^6^0*3^5^0+2^8^0*5^3^0

Explanation:

With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:

Numerator

48^5^0+80^3^0=(2^4*3)^5^0+(2^4*5)^3^0

Continuing the simplification:

(2^4*3)^5^0+(2^4*5)^3^0=2^2^0^0*3^5^0+2^1^2^0*5^3^0

Now, these factors have in common a 2^1^2^0 . Factor this out:

2^1^2^0(2^8^0*3^5^0+5^3^0)

Denominator

This is much simpler:

4^2^0=(2^2)^2^0=2^4^0

Now, return to your fraction:

$\frac{2^1^2^0(2^8^0*3^5^0+5^3^0)}{2^4^0}$

Cancel out the common factors of :

$\frac{2^1^2^0(2^8^0*3^5^0+5^3^0)}{2^4^0}=2^8^0(2^8^0*3^5^0+5^3^0)=2^1^6^0*3^5^0+2^8^0*5^3^0$

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Example Question #1 : How To Find Patterns In Exponents

What digit appears in the units place when $2^{102}$ is multiplied out?

Possible Answers:

Correct answer:

Explanation:

This problem is quite simple if you recall that the units place of powers of 2 follows a simple 4-step sequence.

Observe the first few powers of 2:

2¹= 2, 2²= 4, 2³= 8, 2⁴= 16, 2⁵= 32, 2⁶= 64, 2⁷= 128, 2⁸= 256 . . .

The units place follows a sequence of 2, 4, 8, 6, 2, 4, 8, 6, etc. Thus, divide 102 by 4. This gives a remainder of 2.

The second number in the sequence is 4, so the answer is 4.

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Example Question #1 : How To Find Patterns In Exponents

Which of the following is a multiple of 9^2 ?

Possible Answers:

945

324

1350

315

264

Correct answer:

324

Explanation:

For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:

9^2=(3^2)^2=3^4

Now, in order for you to have a number that is a multiple of this, you will need to have at least 3^4 in the prime factorization of the given number. For each of the answer choices, you have:

315=3^2*5*7

264=2^3*3*11

324=2^2*3^4 ; This is the answer.

1350=2*3^3*5^2

945=3^3*5*7

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Example Question #10 : Pattern Behaviors In Exponents

Simplify the following:

$\frac{25^4^4-5^8^6}{12}$

Possible Answers:

2*5^8^6

2*5^2^3

$\frac{4}{5^2}$

$\frac{25^2}{24}$

12*5^2

Correct answer:

2*5^8^6

Explanation:

Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the into its prime factors:

25^4^4-5^8^6=(5^2)^4^4-5^8^6=5^8^8-5^8^6

Note that these have a common factor of 5^8^6 . Therefore, you can rewrite the numerator as:

5^8^6(5^2-1)=5^8^6(25-1)=5^8^6(24)

Now, put this back into your fraction:

$\frac{5^8^6(24)}{12} = 5^8^6(2)=2*5^8^6$

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Example Question #271 : Exponents

$3 ^{x}= 9 ^{y+1}$ .

Express in terms of .

Possible Answers:

$y = \frac{1}{2}x- \frac{1}{2}$

y = 2x-2

$y = \frac{1}{2}x- 1$

y = 2x-1

$y = x- \frac{1}{2}$

Correct answer:

$y = \frac{1}{2}x- 1$

Explanation:

$3 ^{x}= 9 ^{y+1}$

$3 ^{x}= (3 ^{2} )^{y+1}$

$3 ^{x}= 3 ^{2(y+1)} = 3 ^{2y+2}$

x = 2y + 2

2y = x-2

$y = \frac{x-2}{2} = \frac{1}{2}x- 1$

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Example Question #272 : Exponents

$10,000^{3x+7} = 10 ^{y- 2}$

Express in terms of .

Possible Answers:

y = 3x+9

$y = \frac{1}{12}x+ 30$

y = 12x+ 36

y = 12x+ 30

$y = \frac{1}{12}x+ 26$

Correct answer:

y = 12x+ 30

Explanation:

$10,000^{3x+7} = 10 ^{y- 2}$

$\left ( 10 ^{4} \right )^{3x+7} = 10 ^{y- 2}$

$10 ^{4 \left ( 3x+7 \right )} = 10 ^{y- 2}$

$10 ^{12x+28} = 10 ^{y- 2}$

y - 2 = 12x + 28

y = 12x + 28+ 2 =12x+ 30

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Example Question #273 : Exponents

$16^{5x-17} = 8^{2y + 1}$

Express in terms of .

Possible Answers:

$y = \frac{10}{3}x- 10$

y = 10x-71

y = 10x-65

$y = \frac{10}{3}x- \frac{71}{6}$

$y = \frac{10}{3}x- \frac{65}{6}$

Correct answer:

$y = \frac{10}{3}x- \frac{71}{6}$

Explanation:

$16^{5x-17} = 8^{2y + 1}$

$\left (2^{4} \right )^{5x-17} =( 2 ^{3})^{2y + 1}$

$2^{4(5x-17) }= 2 ^{3(2y + 1)}$

$2^{20x-68}= 2 ^{6y + 3}$

6y+3 = 20x-68

6y+3 -3= 20x-68 - 3

6y = 20x-71

$\frac{6y}{6} = \frac{20x-71}{6}$

$y = \frac{10}{3}x- \frac{71}{6}$

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SAT Math : Pattern Behaviors in Exponents

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #268 : Exponents

Example Question #269 : Exponents

Example Question #12 : Exponents

Example Question #1 : How To Find Patterns In Exponents

Example Question #1 : How To Find Patterns In Exponents

Example Question #1 : How To Find Patterns In Exponents

Example Question #10 : Pattern Behaviors In Exponents

Example Question #271 : Exponents

Example Question #272 : Exponents

Example Question #273 : Exponents

All SAT Math Resources

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