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SAT Math : Algebra

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Example Questions

Example Question #2 : How To Find Patterns In Exponents

Simplify the following:

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

Possible Answers:

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

12^3^0+5^1^0

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

17^8^0

17^5^0

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 #201 : 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 #22 : Algebra

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

Possible Answers:

264

1350

324

315

945

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 #202 : Exponents

Simplify the following:

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

Possible Answers:

2*5^2^3

12*5^2

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

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

2*5^8^6

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 #685 : Algebra

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

Express in terms of .

Possible Answers:

y = 2x-1

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

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

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

y = 2x-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 #686 : Algebra

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

Express in terms of .

Possible Answers:

y = 3x+9

y = 12x+ 30

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

y = 12x+ 36

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

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 #22 : How To Find Patterns In Exponents

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

Express in terms of .

Possible Answers:

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

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

y = 10x-65

y = 10x-71

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

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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Example Question #688 : Algebra

$5^{x} \cdot 25 ^{y} = 5 ^{y} \cdot 125 ^{x}$

Express in terms of .

Possible Answers:

y = x+ 1

y = x+2

y = 3x

y = 2x

y =x+ 3

Correct answer:

y = 2x

Explanation:

To solve this problem, first convert all numbers to have the like base of five.

$5^{x} \cdot 25 ^{y} = 5 ^{y} \cdot 125 ^{x}$

$5^{x} \cdot\left ( 5 ^{2} \right ) ^{y} = 5 ^{y} \cdot\left ( 5 ^{3} \right ) ^{x}$

$5^{x} \cdot 5 ^{2y} = 5 ^{y} \cdot 5 ^{3x}$

Recall the rule of exponents which states, when multiplying like bases we add their exponents.

$5^{x+2y} = 5 ^{3x+ y}$

Since we have the same base we can set the exponents equal to each other and solve.

x+2y = 3x+y

x+2y - x-y = 3x+y - x-y

y = 2x

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Example Question #689 : Algebra

Solve for :

$4^{3t+ 8} = 8 ^{2t- 11}$

Possible Answers:

t = 3

t = 19

t = 6

$t = \frac{9}{2}$

This statement has no solution.

Correct answer:

This statement has no solution.

Explanation:

To solve this problem, first convert all numbers to have the like base of two

$4^{3t+ 8} = 8 ^{2t- 11}$

$\left (2^{2} \right )^{3t+ 8} = \left (2^{3} \right ) ^{2t- 11}$

Recall that when exponents are raised to another power the exponents are multiplied together.

$2^{2(3t+ 8)} = 2^{3(2t- 11)}$

Since we have the same base we can set the exponents equal to each other and solve.

2(3t+ 8) =3(2t- 11)

6t+16 =6t-33

6t+16 - 6t =6t-33 - 6t

16 = -33

This statement is identically false, so the original statement has no solution.

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Example Question #21 : How To Find Patterns In Exponents

Solve for :

$3^{4y-2 } = 27 ^{y}$

Possible Answers:

The equation has no solution.

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

y = -2

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

y = 2

Correct answer:

y = 2

Explanation:

To solve this problem, first convert all numbers to have the like base of three.

$3^{4y-2 } = 27 ^{y}$

$3^{4y-2 } = \left ( 3^{3} \right ) ^{y}$

$3^{4y-2 } = 3^{3y}$

Since we have the same base we can set the exponents equal to each other and solve.

3y= 4y-2

3y - 3y + 2 = 4y-2 - 3y + 2

y = 2

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SAT Math : Algebra

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #2 : How To Find Patterns In Exponents

Example Question #201 : Exponents

Example Question #22 : Algebra

Example Question #202 : Exponents

Example Question #685 : Algebra

Example Question #686 : Algebra

Example Question #22 : How To Find Patterns In Exponents

Example Question #688 : Algebra

Example Question #689 : Algebra

Example Question #21 : How To Find Patterns In Exponents

All SAT Math Resources

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