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# NEW!!! Tough and tricky exponents and roots questions

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NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 02:03
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Exponents and roots problems are very common on the GMAT. So, it's extremely important to know how to manipulate them, how to factor out, take roots, multiply, divide, etc. Below are 11 problems to test your skills. Please post your thought process/solutions along with the answers.

I'll post OA's with detailed solutions tomorrow. Good luck.

1. What is the value of $$\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}$$?
A. $$2\sqrt{5}$$
B. $$\sqrt{55}$$
C. $$2\sqrt{15}$$
D. 50
E. 60

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029216

2. What is the units digit of $$(17^3)^4-1973^{3^2}$$?
A. 0
B. 2
C. 4
D. 6
E. 8

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029219

3. If $$5^{10x}=4,900$$ and $$2^{\sqrt{y}}=25$$ what is the value of $$\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}$$?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029221

4. What is the value of $$5+4*5+4*5^2+4*5^3+4*5^4+4*5^5$$?
A. 5^6
B. 5^7
C. 5^8
D. 5^9
E. 5^10

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029222

5. If $$x=23^2*25^4*27^6*29^8$$ and is a multiple of $$26^n$$, where $$n$$ is a non-negative integer, then what is the value of $$n^{26}-26^n$$?
A. -26
B. -25
C. -1
D. 0
E. 1

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029223

6. If $$x=\sqrt[5]{-37}$$ then which of the following must be true?
A. $$\sqrt{-x}>2$$
B. x>-2
C. x^2<4
D. x^3<-8
E. x^4>32

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029224

7. If $$x=\sqrt{10}+\sqrt[3]{9}+\sqrt[4]{8}+\sqrt[5]{7}+\sqrt[6]{6}+\sqrt[7]{5}+\sqrt[8]{4}+\sqrt[9]{3}+\sqrt[10]{2}$$, then which of the following must be true:
A. x<6
B. 6<x<8
C. 8<x<10
D. 10<x<12
E. x>12

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029227

8. If $$x$$ is a positive number and equals to $$\sqrt{6+{\sqrt{6+\sqrt{6+\sqrt{6+...}}}}}$$, where the given expression extends to an infinite number of roots, then what is the value of x?
A. $$\sqrt{6}$$
B. 3
C. $$1+\sqrt{6}$$
D. $$2\sqrt{3}$$
E. 6

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029228

9. If $$x$$ is a positive integer then the value of $$\frac{22^{22x}-22^{2x}}{11^{11x}-11^x}$$ is closest to which of the following?
A. $$2^{11x}$$
B. $$11^{11x}$$
C. $$22^{11x}$$
D. $$2^{22x}*11^{11x}$$
E. $$2^{22x}*11^{22x}$$

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029229

10. Given that $$5x=125-3y+z$$ and $$\sqrt{5x}-5-\sqrt{z-3y}=0$$, then what is the value of $$\sqrt{\frac{45(z-3y)}{x}}$$?
A. 5
B. 10
C. 15
D. 20
E. Can not be determined

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029231

11. If $$x>0$$, $$x^2=2^{64}$$ and $$x^x=2^y$$ then what is the value of $$y$$?
A. 2
B. 2^(11)
C. 2^(32)
D. 2^(37)
E. 2^(64)

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029232
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 11:34
1
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4. What is the value of 5+4*5+4*5^2+4*5^3+4*5^4+4*5^5?
A. 5^6
B. 5^7
C. 5^8
D. 5^9
E. 5^10

a(r^n - 1) / (r-1)
5+4*5+4*5^2+4*5^3+4*5^4+4*5^5
1+4+4*5+4*5^2+4*5^3+4*5^4+4*5^5
1 + 4(5^6 -1)/4
5^6

Option
[Reveal] Spoiler:
A
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 11:36
1
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5. If x=23^2*25^4*27^6*29^8 and is a multiple of 26^n, where n is a non-negative integer, then what is the value of n^26-26^n?
A. -26
B. -25
C. -1
D. 0
E. 1

n=0 because x is not a multiple of 26

n^26 - 26^n = 0-1 = -1

Option
[Reveal] Spoiler:
C
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 11:44
2
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6. If x=\sqrt[5]{-37} then which of the following must be true?
A. \sqrt{-x}>2
B. x>-2
C. x^2<4
D. x^3<-8
E. x^4>32

x^5 = -37

-2^5 = -32
-3<x<-2

A. Eliminated
B. Eliminated
C. x^2<4 = -2<x<2 Eliminated
D. x^3<-8 Correct

Option
[Reveal] Spoiler:
D
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 11:51
1
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8. If x is a positive number and equals to \sqrt{6+{\sqrt{6+\sqrt{6+\sqrt{6+...}}}}}, where the given expression extends to an infinite number of roots, then what is the value of x?
A. \sqrt{6}
B. 3
C. 1+\sqrt{6}
D. 2\sqrt{3}
E. 6

x= √(6+√(6+√(6+...
x^2 = 6 + x
x^2-x-6=0
(x-3)(x+2) = 0
x = 3 or -2
Value of sqrt cant be -ve
Option
[Reveal] Spoiler:
B
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:01
10. Given that 5x=125-3y+z and \sqrt{5x}-25-\sqrt{z-3y}=0, then what is the value of \sqrt{\frac{45(z-3y)}{x}}?
A. 5
B. 10
C. 15
D. 20
E. Can not be determined

5x-125=z-3y

sqrt{5x}-25=\sqrt{z-3y}
(sqrt{5x}-25)^2=z-3y

(sqrt{5x}-25)^2=5x-125
\sqrt{5x}=15
x=5*9=45

so the expression \sqrt{\frac{45(z-3y)}{x} = \sqrt{45(15-25)/45}=\sqrt{-10}

E is the answ
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:07
7. If x=\sqrt{10}+\sqrt[3]{9}+\sqrt[4]{8}+\sqrt[5]{7}+\sqrt[6]{6}+\sqrt[7]{5}+\sqrt[8]{4}+\sqrt[9]{3}+\sqrt[10]{2}, then which of the following must be true:
A. x<6
B. 6<x<8
C. 8<x<10
D. 10<x<12
E. x>12

my answ to this q is intuitive.

everyone knows that sqroot10=3.16
\sqrt[3]{9} is approx \sqrt[3]{2^3} >2 (but less than 3)
the rest of roots are more than 1 and less than 2

so we have 3.16+2,..+1+1+1+1+1+1+1 = approx 12.16

E is the answ
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:17
9. If x is a positive integer then the value of \frac{22^{22x}-22^{2x}}{11^{11x}-11^x} is closest to which of the following?
A. 2^{11x}
B. 11^{11x}
C. 22^{11x}
D. 2^{22x}*11^{11x}
E. 2^{22x}*11^{22x}

let 11=a

so we have

(2a^(2ax)-2a^2x)/(a^(ax)-a^x)=2*((a^(ax)-a^x)*(a^(ax)+a^x))/(a^(ax)-a^x)=2(a^(ax)+a^x)

2(a^(ax)+a^x)=2*(11^(11x)+11^x)
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:18
1
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Expert's post
Bunuel wrote:

3. If $$5^{10x}=4,900$$ and $$2^{\sqrt{y}}=25$$ what is the value of $$\frac{5^{(x-1)^5}}{4^{-\sqrt{y}}$$?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

5. If $$x=23^2*25^4*27^6*29^8$$ and is a multiple of $$26^n$$, where $$n$$ is a non-negative integer, then what is the value of $$n^26-26^n$$?
A. -26
B. -25
C. -1
D. 0
E. 1

Welcome back Bunuel!

Some nice questions. A couple of comments:

Q5 has a typo at the end: $$n^26-26^n$$. I think you need to include the exponent 26 in braces {26} to get $$n^{26}-26^n$$

In Q3, unless I'm making some kind of stupid mistake, the value of the expression in the question is close to 600 (so isn't among the answers). I'm wondering if you meant the question to instead read something more like:

3. If $$5^{10x}=4,900$$ and $$4^{\sqrt{y}}=25$$ what is the value of $$\frac{5^{5x-1}}{2^{\sqrt{y}}$$?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

In that case, I get one of the five answer choices (A).
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:27
1
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Expert's post
LalaB wrote:
9. If x is a positive integer then the value of \frac{22^{22x}-22^{2x}}{11^{11x}-11^x} is closest to which of the following?
A. 2^{11x}
B. 11^{11x}
C. 22^{11x}
D. 2^{22x}*11^{11x}
E. 2^{22x}*11^{22x}

let 11=a

so we have

(2a^(2ax)-2a^2x)/(a^(ax)-a^x)=2*((a^(ax)-a^x)*(a^(ax)+a^x))/(a^(ax)-a^x)=2(a^(ax)+a^x)

2(a^(ax)+a^x)=2*(11^(11x)+11^x)

And it looks as though every other question has been solved except for this one. I won't solve it completely, but I'll help people get started. Here, since we are asked for an approximation only, we don't need to compute an exact value. You might notice that if x is a positive integer, $$22^{22x}$$ is vastly bigger than $$22^{2x}$$. Since that's true, if we only need to estimate, we can ignore the small term. We can do the same thing in the denominator.
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:40
yep, if the 3d q is -If 5^{10x}=4,900 and 4^{\sqrt{y}}=25 what is the value of \frac{5^{5x-1}}{2^{\sqrt{y}}?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14""

then

5^{10x}=70^2
(5^{5x})2=70^2
5^{5x}=70

4^{\sqrt{y}}=25
2^{\sqrt{y}}=5

70/5*5=14/5 ans is A
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 12:48
IanStewart wrote:

And it looks as though every other question has been solved except for this one. I won't solve it completely, but I'll help people get started. Here, since we are asked for an approximation only, we don't need to compute an exact value. You might notice that if x is a positive integer, $$22^{22x}$$ is vastly bigger than $$22^{2x}$$. Since that's true, if we only need to estimate, we can ignore the small term. We can do the same thing in the denominator.

2a^(2ax)/a^(ax)=2*a^(ax)=2*11^(11x)

ans is
[Reveal] Spoiler:
B
?
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 13:31
1
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Expert's post
LalaB wrote:
IanStewart wrote:

And it looks as though every other question has been solved except for this one. I won't solve it completely, but I'll help people get started. Here, since we are asked for an approximation only, we don't need to compute an exact value. You might notice that if x is a positive integer, $$22^{22x}$$ is vastly bigger than $$22^{2x}$$. Since that's true, if we only need to estimate, we can ignore the small term. We can do the same thing in the denominator.

2a^(2ax)/a^(ax)=2*a^(ax)=2*11^(11x)

ans is
[Reveal] Spoiler:
B
?

Your substitution might be making things more difficult. If we want to estimate the value of

$$\frac{22^{22x}-22^{2x}}{11^{11x}-11^x}$$

then we can ignore the comparatively small values in the numerator and denominator; this is roughly equal to

$$\frac{22^{22x}}{11^{11x}}$$

which is equal to

$$\frac{22^{22x}}{11^{11x}} = \frac{2^{22x} 11^{22x}}{11^{11x}} = 2^{22x} 11^{11x}$$
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 17:00
IanStewart wrote:
Bunuel wrote:

3. If $$5^{10x}=4,900$$ and $$2^{\sqrt{y}}=25$$ what is the value of $$\frac{5^{(x-1)^5}}{4^{-\sqrt{y}}$$?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

5. If $$x=23^2*25^4*27^6*29^8$$ and is a multiple of $$26^n$$, where $$n$$ is a non-negative integer, then what is the value of $$n^26-26^n$$?
A. -26
B. -25
C. -1
D. 0
E. 1

Welcome back Bunuel!

Some nice questions. A couple of comments:

Q5 has a typo at the end: $$n^26-26^n$$. I think you need to include the exponent 26 in braces {26} to get $$n^{26}-26^n$$

In Q3, unless I'm making some kind of stupid mistake, the value of the expression in the question is close to 600 (so isn't among the answers). I'm wondering if you meant the question to instead read something more like:

3. If $$5^{10x}=4,900$$ and $$4^{\sqrt{y}}=25$$ what is the value of $$\frac{5^{5x-1}}{2^{\sqrt{y}}$$?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

In that case, I get one of the five answer choices (A).

Hello Ian. Glad to be back!

For Q5: included the exponent 26 in braces {26} to avoid confusion;
For Q3: added the brackets which were lost in copy/past, it should read $$\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}$$.
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 20:22
Hi,
1 - C
2 - E
3 - E
4 - A
5 -
6 - B
7 - E
8 - B
9 -
10 - E
11 - E

Unable to solve for 5th & 9th
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 13:39
LalaB wrote:
2. What is the units digit of (17^3)^4-1973^{3^2}?
A. 0
B. 2
C. 4
D. 6
E. 8

7^12-3^9
11-3=8

ANS IS E

Bunuel wrote:
2. What is the units digit of $$(17^3)^4-1973^{3^2}$$?
A. 0
B. 2
C. 4
D. 6
E. 8

i believe the answer is : 7^12-3^6 ---> 11-9= 2 ---> B

note: I believe the part I wrote (which i colored red) is wrong. Can someone explain why? when an exponent is raised to another exponent then what operation should we do first? - Okey! Just read the rules about routs that Bunuel posted and found my mistake! (exponentiation by stacked symbols)

Last edited by SonyGmat on 13 Jan 2012, 23:11, edited 6 times in total.
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 15:09
LalaB wrote:
10. Given that 5x=125-3y+z and \sqrt{5x}-25-\sqrt{z-3y}=0, then what is the value of \sqrt{\frac{45(z-3y)}{x}}?
A. 5
B. 10
C. 15
D. 20
E. Can not be determined

5x-125=z-3y

sqrt{5x}-25=\sqrt{z-3y}
(sqrt{5x}-25)^2=z-3y

(sqrt{5x}-25)^2=5x-125
\sqrt{5x}=15
x=5*9=45

so the expression \sqrt{\frac{45(z-3y)}{x} = \sqrt{45(15-25)/45}=\sqrt{-10}

E is the answ

bunuel wrote:
10. Given that $$5x=125-3y+z$$ and $$\sqrt{5x}-25-\sqrt{z-3y}=0$$, then what is the value of $$\sqrt{\frac{45(z-3y)}{x}}$$?
A. 5
B. 10
C. 15
D. 20
E. Can not be determined

We know that:

$$5x=125-3y+z$$ ---> [highlight]$$z-3y=5x-125$$ (1)[/highlight]

and that:

$$\sqrt{5x}-25-\sqrt{z-3y}=0$$ ---> $$\sqrt{5x}-25=\sqrt{z-3y}$$ ----> $$[\sqrt{5x}-25]^2=[\sqrt{z-3y}]^2$$ ----> using (1): $$[\sqrt{5x}-25]^2=[\sqrt{5x-125}]^2$$ --->

$$5x-50\sqrt{5x}+625=5x-125$$ ---> $$50\sqrt{5x}=750$$ ---> $$\sqrt{5x}=15$$ ---> $$5x=15*15$$--->[[highlight]$$x=45$$ (2)[/highlight]

(1), (2) ---> $$z-3y=5*45-125$$--->[highlight]$$z-3y=100$$ (3)[/highlight]

therefore: $$\sqrt{\frac{45(z-3y)}{x}}$$ ---->using (2),(3) :$$\sqrt{\frac{45*100}{45}}$$ ---> 10 ---->B
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 18:42
1
KUDOS
IanStewart wrote:
Bunuel wrote:

3. If $$5^{10x}=4,900$$ and $$2^{\sqrt{y}}=25$$ what is the value of $$\frac{5^{(x-1)^5}}{4^{-\sqrt{y}}$$?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

5. If $$x=23^2*25^4*27^6*29^8$$ and is a multiple of $$26^n$$, where $$n$$ is a non-negative integer, then what is the value of $$n^26-26^n$$?
A. -26
B. -25
C. -1
D. 0
E. 1

Welcome back Bunuel!

Some nice questions. A couple of comments:

Q5 has a typo at the end: $$n^26-26^n$$. I think you need to include the exponent 26 in braces {26} to get $$n^{26}-26^n$$

In Q3, unless I'm making some kind of stupid mistake, the value of the expression in the question is close to 600 (so isn't among the answers). I'm wondering if you meant the question to instead read something more like:

3. If $$5^{10x}=4,900$$ and $$4^{\sqrt{y}}=25$$ what is the value of $$\frac{5^{5x-1}}{2^{\sqrt{y}}$$?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

In that case, I get one of the five answer choices (A).

The answer should be E. 14

Here is how:

$$5^{10x}=4900$$
so $${(5^{5x})}^2=70^2$$

Take square root of both sides >

so $$5^{5x}=70$$

Also from question stem:

$$2^{\sqrt{y}} = 5^2$$

So (from the question stem)

$$4^{-\sqrt{y}} = \frac{1}{4^{\sqrt{y}}}$$
$$4^{-\sqrt{y}} = (\frac{1}{2^{\sqrt{y}}})^2 = \frac{1}{5^4}$$

Now $$5^{(x-1)^5}$$

Should be $$\frac{5^{5x}}{5^5}$$

We know $$5^{10x}=4900$$

so $$5^{10x}=70^2$$

so $$5^{5x}=70$$

So $$\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}$$

is basically $$\frac{5^x}{5^5}*5^4$$

We already know the value of $$5^x$$ which is $$70$$

So now it becomes $$\frac{70}{5^5}*5^4$$

Which should resolve to $$14$$

Hence Answer = E = $$14$$
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 20:05
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Q1.

$$\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}$$

First look and we know that this is of the type:

let a = $$\sqrt{25+10\sqrt{6}}$$

Let b = $$\sqrt{25-10\sqrt{6}}$$

let $$a+b=x$$

Then $$(a+b)^2=x^2$$

So $$a^2+b^2+2ab=x^2$$

So $$a^2 = {25+10\sqrt{6}}$$
So $$b^2 = {25-10\sqrt{6}}$$
So $$a^2+b^2=50$$

Now on to $$2ab$$

$$a=\sqrt{25+10\sqrt{6}}$$
$$b=\sqrt{25-10\sqrt{6}}$$

let $$ab=y$$

then $$\sqrt{25+10\sqrt{6}}*\sqrt{25-10\sqrt{6}}=y$$
So $$({25+10\sqrt{6}})*({25-10\sqrt{6}})=y^2$$
so $$625-600=y^2$$
so $$y=5$$
so $$2ab=2y=10$$
so $$x^2=a^2+b^2+2ab=50+10=60$$
so $$x^2=60$$
so $$x=\sqrt{4*15}$$
so $$x=2\sqrt{15}$$

Hence Answer is C. $$2\sqrt{15}$$
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 20:35
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Q2. What is the units digit of $$(17^3)^4-1973^{3^2}$$?[/b]

Now we know that:

$$(17^3)^4 = (17^6)^2$$

So the expression now becomes

$$(17^6)^2 - (1973^3)^2$$

Now we know that $$a^2-b^2$$ can be written as $$(a+b)*(a-b)$$

Let $$a=17^6$$ and $$b=1973^3$$

Now on to $$(a+b)*(a-b)$$

$$(17^6+1973^3)*(17^6-1973^3)$$

Lets see what the units digit of $$17^6$$ is:
We are only concerned with $$7$$:
$$7*7*7*7*7*7= ..........9$$ unit digit of 9

Lets see what the units digit of $$1973^3$$ is:
We are only concerned with $$3$$:
$$3*3*3= ..........7$$ unit digit of 7

Lets translate this into our expression for $$(a+b)*(a-b)$$

$$(17^6+1973^3)*(17^6-1973^3)$$

Should in digit terms give:

$$(9+7)*(9-7)$$
$$(6)*(2)$$

Should Yield a units digit of 2.

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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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13 Jan 2012, 20:45
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Q3.

$$5^{10x}=4900$$
so $${(5^{5x})}^2=70^2$$

Take square root of both sides >

so $$5^{5x}=70$$

Also from question stem:

$$2^{\sqrt{y}} = 5^2$$

So (from the question stem)

$$4^{-\sqrt{y}} = \frac{1}{4^{\sqrt{y}}}$$
$$4^{-\sqrt{y}} = (\frac{1}{2^{\sqrt{y}}})^2 = \frac{1}{5^4}$$

Now $$5^{(x-1)^5}$$

Should be $$\frac{5^{5x}}{5^5}$$

We know $$5^{10x}=4900$$

so $$5^{10x}=70^2$$

so $$5^{5x}=70$$

So $$\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}$$

is basically $$\frac{5^x}{5^5}*5^4$$

We already know the value of $$5^x$$ which is $$70$$

So now it becomes $$\frac{70}{5^5}*5^4$$

Which should resolve to $$14$$

Hence Answer = E = $$14$$
_________________

"Nowadays, people know the price of everything, and the value of nothing." Oscar Wilde

Re: NEW!!! Tough and tricky exponents and roots questions   [#permalink] 13 Jan 2012, 20:45

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