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Math Expert V
Joined: 02 Sep 2009
Posts: 58325
Exponents and Roots on the GMAT: Tips and hints  [#permalink]

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44

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 $$\sqrt{x}$$, 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, $$\sqrt{125} =5$$ and $$\sqrt{-64} =-4$$.

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

Theory: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

All DS roots problems to practice: search.php?search_id=tag&tag_id=49
All PS roots problems to practice: search.php?search_id=tag&tag_id=113

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Please share your Exponents and Roots tips below and get kudos point. Thank you.
_________________
Intern  Joined: 27 Aug 2014
Posts: 27
GMAT Date: 09-27-2014
Re: Exponents and Roots on the GMAT: Tips and hints  [#permalink]

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1
3^n will always have an even number of tens.

Example: 27, 81, 729, etc
VP  D
Joined: 09 Mar 2016
Posts: 1230
Re: Exponents and Roots on the GMAT: Tips and hints  [#permalink]

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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 $$\sqrt{x}$$, 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, $$\sqrt{125} =5$$ and $$\sqrt{-64} =-4$$.

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

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

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

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

Tough and tricky DS exponents and roots questions with detailed solutions: http://gmatclub.com/forum/tough-and-tri ... 25967.html
Tough and tricky PS exponents and roots questions with detailed solutions: http://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) Math Expert V
Joined: 02 Sep 2009
Posts: 58325
Re: Exponents and Roots on the GMAT: Tips and hints  [#permalink]

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1
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 $$\sqrt{x}$$, 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, $$\sqrt{125} =5$$ and $$\sqrt{-64} =-4$$.

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

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

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

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

Tough and tricky DS exponents and roots questions with detailed solutions: http://gmatclub.com/forum/tough-and-tri ... 25967.html
Tough and tricky PS exponents and roots questions with detailed solutions: http://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 $$(5^3)^2$$
_________________
Intern  B
Joined: 02 Jun 2013
Posts: 44
Re: Exponents and Roots on the GMAT: Tips and hints  [#permalink]

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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
_________________
Nothing comes easy ! Neither do i want !!
Math Expert V
Joined: 02 Sep 2009
Posts: 58325
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;
$$\sqrt{16} = 2$$, 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]

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_________________ Re: Exponents and Roots on the GMAT: Tips and hints   [#permalink] 03 May 2019, 09:37
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