from: category_eng

1.

Manipulation

2.

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What is the value of $frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?$

$extbf{(A)} -1 qquad extbf{(B)} 1 qquad extbf{(C)} frac{5}{3} qquad extbf{(D)} 2013 qquad extbf{(E)} 2^{4024}$

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3.

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Find the value of $x$ that satisfies the equation $25^{-2} = frac{5^{48/x}}{5^{26/x} cdot 25^{17/x}}.$

$extbf{(A) } 2 qquad extbf{(B) } 3 qquad extbf{(C) } 5 qquad extbf{(D) } 6 qquad extbf{(E) } 9$

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4.

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Suppose that $4^a = 5$ , $5^b = 6$ , $6^c = 7$ , and $7^d = 8$ . What is $a cdot bcdot c cdot d$ ?

$mathrm{(A)} 1 qquad mathrm{(B)} frac{3}{2} qquad mathrm{(C)} 2 qquad mathrm{(D)} frac{5}{2} qquad mathrm{(E)} 3$

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5.

Case Devision

6.

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What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$ ?

$mathrm{(A) } frac17qquad mathrm{(B) } frac27qquad mathrm{(C) } 1qquad mathrm{(D) } 7qquad mathrm{(E) } 1...$

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7.

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Real numbers $a$ and $b$ satisfy the equations $3^{a} = 81^{b + 2}$ and $125^{b} = 5^{a - 3}$ . What is $ab$ ?

$ext{(A)} -60 qquad ext{(B)} -17 qquad ext{(C)} 9 qquad ext{(D)} 12 qquad ext{(E)} 60$

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8.

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1. 4	Complex

2.

Factoring out, we get: $frac{2^{2012}(2^2 + 1)}{2^{2012}(2^2-1)} ?$

Cancelling out the $2^{2012}$ from the numerator and denominator, we see that it simplifies to $oxed{ extbf{(C) }frac{5}{3}}$ .

3.

Manipulate the powers of $5$ in order to get a clean expression.

$frac{5^{frac{48}{x}}}{5^{frac{26}{x}} cdot 25^{frac{17}{x}}} = frac{5^{frac{48}{x}}}{5^{frac{26}{x}} cdot 5^{frac{3...$
$25^{-2} = (5^2)^{-2} = 5^{-4}$
$5^{-4} = 5^{-frac{12}{x}}$

If two numbers are equal, and their bases are equal, then their exponents are equal as well. Set the two exponents equal to each other.

$egin{align*}-4&=frac{-12}{x}-4x&=-12x&=oxed{ extbf{(B) } 3}end{align*}$

4.

$8=7^d$ $8=left(6^c<br /> ight)^d$ $8=left(left(5^b<br /> ight)^c<br /> ight)^d$ $8=left(left(left(4^a<br /> ight)^b<br /> ight)^c<br /> ight)^d$ $8=4^{acdot bcdot ccdot d}$ $2^3=2^{2cdot acdot bcdot ccdot d}$ $3=2cdot acdot bcdot ccdot d$ $acdot bcdot ccdot d=oxed{mathrm{(B)} dfrac{3}{2}}$

Solution using logarithms

We can write $a$ as $log_4 5$ , $b$ as $log_56$ , $c$ as $log_67$ , and $d$ as $log_78$ .
We know that $log_b a$ can be rewritten as $frac{log a}{log b}$ , so $a*b*c*d=$
$frac{log5}{log4}cdotfrac{log6}{log5}cdotfrac{log7}{log6}cdotfrac{log8}{log7}$

$frac{log8}{log4}$

$frac{3log2}{2log2}$

$oxed{frac{3}{2}}$

5. 9	Understanding

6.

Taking the seventh root of both sides, we get $(7x)^2=14x$ .

Simplifying the LHS gives $49x^2=14x$ , which then simplifies to $7x=2$ .

Thus, $x=frac{2}{7}$ , and the answer is $mathrm{(B)}$ .

7.

$81^{b+2} = 3^{4(b+2)} = 3^a Longrightarrow a = 4b+8$

And

$125^{b} = 5^{3b} = 5^{a-3} Longrightarrow a - 3 = 3b$

Substitution gives $4b+8 - 3 = 3b Longrightarrow b = -5$ , and solving for $a$ yields $-12$ . Thus $ab = 60 mathrm{(E)}$ .

8.	Calculation
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