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Exponents and Roots on the GMAT: Tips and hints : Quantitative Questions
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
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dave13 wrote:
Bunuel wrote:

Exponents and Roots: Tips and hints



!
This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

DEFINITION - 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.

TIPS - EXPONENTS

1. 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.

2. 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.

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

4. Negative powers:
\(a^{-n}=\frac{1}{a^n}\)
Important: you cannot rise 0 to a negative power because you get division by 0, which is NOT allowed. For example, \(0^{-1} = \frac{1}{0}=undefined\).

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

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

6. 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)

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

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

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

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

DEFINITION - ROOTS

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

TIPS - ROOTS

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

2. \((\sqrt{x})^n=\sqrt{x^n}\)

3. \(x^{\frac{1}{n}}=\sqrt[n]{x}\)

4. \(x^{\frac{n}{m}}=\sqrt[m]{x^n}\)

5. \({\sqrt{a}}+{\sqrt{b}}\neq{\sqrt{a+b}}\)

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

7. 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.

8. Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).

This week's PS question
This week's DS Question

Theory: https://gmatclub.com/forum/math-number-theory-88376.html

All DS Exponents questions to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=39
All PS Exponents questions to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=60

All DS roots problems to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=49
All PS roots problems to practice: https://gmatclub.com/forum/search.php?se ... tag_id=113

Tough and tricky DS exponents and roots questions with detailed solutions: https://gmatclub.com/forum/tough-and-tri ... 25967.html
Tough and tricky PS exponents and roots questions with detailed solutions: https://gmatclub.com/forum/tough-and-tri ... 25956.html


Please share your Exponents and Roots tips below and get kudos point. Thank you.



What does it mean ? \((a^m)^n\) (if exponentiation is indicated by stacked symbols, the rule is to work from the top down) :?


It means that \(5^{3^2}=5^{(3^2)}=5^9\) and not \((5^3)^2\)
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
HI Bunuel

I get confuse when i should consider both the sign of roots and when not. Please pardon my ignorance, but somehow my concepts are not clear on this .

e.g x^2 = 25 ==> x= +5 or -5
(25)^1/2 ==> 5
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
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rahul2013 wrote:
HI Bunuel

I get confuse when i should consider both the sign of roots and when not. Please pardon my ignorance, but somehow my concepts are not clear on this .

e.g x^2 = 25 ==> x= +5 or -5
(25)^1/2 ==> 5


\(\sqrt{...}\) is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign (\(\sqrt{...}\)) always means non-negative square root.


The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:
When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
Hi Bunuel - thanks a lot for this theory. Just one question - for the negative exponents it does not matter if the base is even or odd right? The result will always be positive, except when the base itself is negative?
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
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davidbeckham wrote:
Hi Bunuel - thanks a lot for this theory. Just one question - for the negative exponents it does not matter if the base is even or odd right? The result will always be positive, except when the base itself is negative?


The result will be positive also when the negative exponent is even. For example, (-2)^(-2) = 1/4.
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
And if both the base and the exponent are odd and negative?

Bunuel wrote:
davidbeckham wrote:
Hi Bunuel - thanks a lot for this theory. Just one question - for the negative exponents it does not matter if the base is even or odd right? The result will always be positive, except when the base itself is negative?


The result will be positive also when the negative exponent is even. For example, (-2)^(-2) = 1/4.
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
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davidbeckham wrote:
And if both the base and the exponent are odd and negative?

Bunuel wrote:
davidbeckham wrote:
Hi Bunuel - thanks a lot for this theory. Just one question - for the negative exponents it does not matter if the base is even or odd right? The result will always be positive, except when the base itself is negative?


The result will be positive also when the negative exponent is even. For example, (-2)^(-2) = 1/4.


The result will be negative. For example, (-2)^(-3)= -1/8.
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
I have a question about notation when powers are raised to powers.

What is the difference between these expressions?

A: 2^2^2
B: 2^(2)^2
C: (2^2)^2

In A, I would say it is the same as 2^(2^2) = 2^4.

In B, it will also be 2^(2^2) = 2^4.

In C, on the other hand, it will be 4^2.

Actually, Im just not sure about A. If the expression is naked (without parenthesis), then how is it treated?

Thanks in advance.
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
A: 2^2^2
B: 2^(2)^2
C: \((2^2)^2\)

A and B are the same. For both A and B, you would compute the topmost exponents first as these are stacked.(tip#6 Operations involving the same exponents)

For C, compute the numbers in the brackets first and then square it. 4^2
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
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Bambi2021 wrote:
I have a question about notation when powers are raised to powers.

What is the difference between these expressions?

A: 2^2^2
B: 2^(2)^2
C: (2^2)^2

In A, I would say it is the same as 2^(2^2) = 2^4.

In B, it will also be 2^(2^2) = 2^4.

In C, on the other hand, it will be 4^2.

Actually, Im just not sure about A. If the expression is naked (without parenthesis), then how is it treated?

Thanks in advance.


In general, in the absence of parentheses, nested operations work from the inside out. Therefore, a^b^c^d = a^(b^(c^d)).
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
Bunuel wrote:

Exponents and Roots: Tips and hints



!
This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

DEFINITION - 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.

TIPS - EXPONENTS

1. 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.

2. 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.

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

4. Negative powers:
\(a^{-n}=\frac{1}{a^n}\)
Important: you cannot rise 0 to a negative power because you get division by 0, which is NOT allowed. For example, \(0^{-1} = \frac{1}{0}=undefined\).

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

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

6. 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)

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

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

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

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

DEFINITION - ROOTS

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

TIPS - ROOTS

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

2. \((\sqrt{x})^n=\sqrt{x^n}\)

3. \(x^{\frac{1}{n}}=\sqrt[n]{x}\)

4. \(x^{\frac{n}{m}}=\sqrt[m]{x^n}\)

5. \({\sqrt{a}}+{\sqrt{b}}\neq{\sqrt{a+b}}\)

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

7. 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.

8. Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).

This week's PS question
This week's DS Question

Theory: https://gmatclub.com/forum/math-number- ... 88376.html

All DS Exponents questions to practice: https://gmatclub.com/forum/search.php?s ... &tag_id=39
All PS Exponents questions to practice: https://gmatclub.com/forum/search.php?s ... &tag_id=60

All DS roots problems to practice: https://gmatclub.com/forum/search.php?s ... &tag_id=49
All PS roots problems to practice: https://gmatclub.com/forum/search.php?s ... tag_id=113

Tough and tricky DS exponents and roots questions with detailed solutions: https://gmatclub.com/forum/tough-and-tr ... 25967.html
Tough and tricky PS exponents and roots questions with detailed solutions: https://gmatclub.com/forum/tough-and-tr ... 25956.html


Please share your Exponents and Roots tips below and get kudos point. Thank you.


Bunuel

Thank you for this helpful guide. For operations involving the same bases, when you have addition or subtraction instead of multiplication or division, the only thing that you can do is to factor out terms to try to simplify, correct?


For example, if you have x^(2+2n)-x^n, you can't add the exponents, but you can try to factor out a term to simplify the expression?
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
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woohoo921 wrote:
Bunuel wrote:

Exponents and Roots: Tips and hints



!
This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

DEFINITION - 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.

TIPS - EXPONENTS

1. 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.

2. 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.

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

4. Negative powers:
\(a^{-n}=\frac{1}{a^n}\)
Important: you cannot rise 0 to a negative power because you get division by 0, which is NOT allowed. For example, \(0^{-1} = \frac{1}{0}=undefined\).

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

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

6. 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)

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

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

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

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

DEFINITION - ROOTS

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

TIPS - ROOTS

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

2. \((\sqrt{x})^n=\sqrt{x^n}\)

3. \(x^{\frac{1}{n}}=\sqrt[n]{x}\)

4. \(x^{\frac{n}{m}}=\sqrt[m]{x^n}\)

5. \({\sqrt{a}}+{\sqrt{b}}\neq{\sqrt{a+b}}\)

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

7. 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.

8. Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).

This week's PS question
This week's DS Question

Theory: https://gmatclub.com/forum/math-number- ... 88376.html

All DS Exponents questions to practice: https://gmatclub.com/forum/search.php?s ... &tag_id=39
All PS Exponents questions to practice: https://gmatclub.com/forum/search.php?s ... &tag_id=60

All DS roots problems to practice: https://gmatclub.com/forum/search.php?s ... &tag_id=49
All PS roots problems to practice: https://gmatclub.com/forum/search.php?s ... tag_id=113

Tough and tricky DS exponents and roots questions with detailed solutions: https://gmatclub.com/forum/tough-and-tr ... 25967.html
Tough and tricky PS exponents and roots questions with detailed solutions: https://gmatclub.com/forum/tough-and-tr ... 25956.html


Please share your Exponents and Roots tips below and get kudos point. Thank you.


Bunuel

Thank you for this helpful guide. For operations involving the same bases, when you have addition or subtraction instead of multiplication or division, the only thing that you can do is to factor out terms to try to simplify, correct?


For example, if you have x^(2+2n)-x^n, you can't add the exponents, but you can try to factor out a term to simplify the expression?

______________________
Pretty much yes.
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
Bunuel wrote:
6. Operations involving the same exponents:
Keep the exponent, multiply or divide the bases

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


Hey avigutman Bunuel - is the above true even if the (Base) itself is negative ?

I dont think so

I am not sure as per screenshot
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
Expert Reply
jabhatta2 wrote:
Bunuel wrote:
6. Operations involving the same exponents:
Keep the exponent, multiply or divide the bases

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


Hey avigutman Bunuel - is the above true even if the (Base) itself is negative ?

I dont think so

I am not sure as per screenshot


It's given there as a general rule, so yes, it's ALWAYS true.


\((-2^3)^2 = (-8)^2 = 64\).

Or since 2 is even, then \((-2^3)^2 = (2^3)^2=8^2=64\).

Or \((-2^3)^2 = (-1*2^3)^2 = (-1)^2*2^6 = 64\).

As I explained here the minus sign just means "multiplied by -1" and the exponent of 2 should be applied to -1 too.

If it were \(-((2^3)^2)\), then \(-((2^3)^2)=-(2^6)=-64\).
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
Expert Reply
jabhatta2 wrote:
Bunuel wrote:
6. Operations involving the same exponents:
Keep the exponent, multiply or divide the bases

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


Hey avigutman Bunuel - is the above true even if the (Base) itself is negative ?

I dont think so

I am not sure as per screenshot

jabhatta2 If a=-2 then that whole thing (-2) is the base for the right side of the equation. a^mn = (-2)^mn

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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
Bunuel wrote:
It's given there as a general rule, so yes, it's ALWAYS true.


\((-2^3)^2 = (-8)^2 = 64\).

Or since 2 is even, then \((-2^3)^2 = (2^3)^2=8^2=64\).

Or \((-2^3)^2 = (-1*2^3)^2 = (-1)^2*2^6 = 64\).

As I explained here the minus sign just means "multiplied by -1" and the exponent of 2 should be applied to -1 too.

If it were \(-((2^3)^2)\), then \(-((2^3)^2)=-(2^6)=-64\).


Hi Bunuel - I agree on the yellow.

However when I do these exponent questions - -- I do it this way instead

I tend to multiply the exponents first (that way, there are fewer exponents)

Example - \((+2^3)^2 \) ------> \( +2^6 \) --------> +64

I am most comfortable doing the above strategy

But when the base is negative, my strategy of multiplying exponents first FAILS

\((-2^3)^2\) -------> \( -2^ 6\) ------------> -64

I know -64 is wrong

I am struggling to see, what is the LOGIC BEHIND why my strategy of multiplying exponents first, fails when the base of the exponent itself is negative

Thank you !
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
avigutman wrote:
jabhatta2 If a=-2 then that whole thing (-2) is the base for the right side of the equation. a^mn = (-2)^mn


Hi avigutman - Once you re-plug a = -2, why not instead ?

a^mn = -2 ^mn

I DONT find it intuitive to have the parenthesis
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Re: Exponents and Roots on the GMAT: Tips and hints [#permalink]
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