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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

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

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

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

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. \(2^{22x}*11^{22x}\)

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

THEORY TO TACKLE THE PROBLEMS ABOVE: For more on number theory check the Number Theory Chapter of Math Book: math-number-theory-88376.html

EXPONENTS

Exponents are a "shortcut" method of showing a number that was multiplied by itself several times. For instance, number \(a\) multiplied \(n\) times can be written as \(a^n\), where \(a\) represents the base, the number that is multiplied by itself \(n\) times and \(n\) represents the exponent. The exponent indicates how many times to multiple the base, \(a\), by itself.

Exponents one and zero: \(a^0=1\) Any nonzero number to the power of 0 is 1. For example: \(5^0=1\) and \((-3)^0=1\) • Note: the case of 0^0 is not tested on the GMAT.

\(a^1=a\) Any number to the power 1 is itself.

Powers of zero: If the exponent is positive, the power of zero is zero: \(0^n = 0\), where \(n > 0\).

If the exponent is negative, the power of zero (\(0^n\), where \(n < 0\)) is undefined, because division by zero is implied.

Powers of one: \(1^n=1\) The integer powers of one are one.

Negative powers: \(a^{-n}=\frac{1}{a^n}\)

Powers of minus one: If n is an even integer, then \((-1)^n=1\).

If n is an odd integer, then \((-1)^n =-1\).

Operations involving the same exponents: Keep the exponent, multiply or divide the bases \(a^n*b^n=(ab)^n\)

\(\frac{a^n}{b^n}=(\frac{a}{b})^n\)

\((a^m)^n=a^{mn}\)

\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\) (if exponentiation is indicated by stacked symbols, the rule is to work from the top down)

Operations involving the same bases: Keep the base, add or subtract the exponent (add for multiplication, subtract for division) \(a^n*a^m=a^{n+m}\)

\(\frac{a^n}{a^m}=a^{n-m}\)

Fraction as power: \(a^{\frac{1}{n}}=\sqrt[n]{a}\)

\(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)

ROOTS

Roots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 and square root of 16=4.

General rules: • \(\sqrt{x}\sqrt{y}=\sqrt{xy}\) and \(\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}\).

• \((\sqrt{x})^n=\sqrt{x^n}\)

• \(x^{\frac{1}{n}}=\sqrt[n]{x}\)

• \(x^{\frac{n}{m}}=\sqrt[m]{x^n}\)

• \({\sqrt{a}}+{\sqrt{b}}\neq{\sqrt{a+b}}\)

• \(\sqrt{x^2}=|x|\), when \(x\leq{0}\), then \(\sqrt{x^2}=-x\) and when \(x\geq{0}\), then \(\sqrt{x^2}=x\)

• When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root.

That is, \(\sqrt{25}=5\), NOT +5 or -5. In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.

• Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).
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Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

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12 Jan 2012, 08:45

for 1: I just read it as (a + b) i. e = \sqrt{(a+b)^2} then substituting the value for a and b a= \sqrt{(25+10[square_root]6)}[/square_root] and b = a= \sqrt{(25-10[square_root]6)}[/square_root]

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

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12 Jan 2012, 08:49

1

This post received KUDOS

wallstreetbarbie wrote:

Can you explain 1? some of the code came out messed up and its difficult to read

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

1st raise it to the 2nd power (in a simple form u have (a+b)^2) (\sqrt{25+ 10[square_root]6}[/square_root]+\sqrt{25- 10[square_root]6}[/square_root] )^2 =

=((\sqrt{25+ 10[square_root]6}[/square_root])^2+ 2((\sqrt{25+ 10[square_root]6}[/square_root])*(\sqrt{25- 10[square_root]6}[/square_root] ) +(\sqrt{25- 10[square_root]6}[/square_root] )^2 (in a simple form u have got (a^2+2ab+b^2)

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

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12 Jan 2012, 10:13

1

This post received KUDOS

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

23^2*25^4*27^6*29^8/26^n= 23^2*25^4*27^6*29^8/(2*13)^n from this point it is obvious, that n =0 (since both 2 and 13 are primes and none of the numbers of numerator can be divisible by 2 and 13)

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

answ is C
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