Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

GMAT: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

The document Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT is a part of the GMAT Course Quantitative Aptitude for GMAT.
All you need of GMAT at this link: GMAT

Exponents: 

Exponents are a "shortcut" method of showing a number that was multiplied by itself several times.

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

  • For instance, number a multiplied n times can be written as an, 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 multiply the base,a, by itself.

Formulas of Exponents

1. Exponents one and zero:

a0=1 Any nonzero number to the power of 0 is 1. 

For example 50=1 and (−3)0=1

Note: the case of 0^0 is not tested on the GMAT. 

a1=a  Any number to the power 1 is itself.

2. Powers of zero: 

If the exponent is positive, the power of zero is zero: 0n=0, where n>0.

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

3. Powers of one: 

1n=1  The integer powers of one are one.

4. Negative powers: 

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Important: you cannot rise 0 to a negative power because you get division by 0, which is NOT allowed. For example, 0−1=1/0=undefined

Try yourself:The value of 2-2 is:
View Solution


5. Powers of minus one: 

If n is an even integer, then (−1)= 1

If n is an odd integer, then (−1)= −1.

6. Operations involving the same exponents: 

Keep the exponent, multiply or divide the bases

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

7. Operations involving the same bases: 

Keep the base, add or subtract the exponent (add for multiplication, subtract for division)

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

8. Fraction as power: 

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT


Try yourself:22 x 23 x 24 is equal to:
View Solution


Roots

Roots (or radicals) are the "opposite" operation of applying exponents. For instance x2=16 and square root of 16=4.

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Formulas of Roots 

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT


Try yourself:What will be the number of zeros in square of 400?
View Solution


Solved Questions on Exponents and Roots 

Section - 1
Simplify the following expressions by combining like terms. If the base is a number, leave the answer in exponential form (i.e. 23, not 8).


Ques 1. x5 x x3
Ans: 
x5 x x3 = x(5+3= x8

Ques 2: 76 x 7
Ans: 
76 x 79 = 7(6+9) = 715

Ques 3: 55/55
Ans: 5(5-3) = 52

Ques 4: (a3)2
Ans:  
(a2)3 = a(3x2) = a6

Ques 5: 4-2 x 4
Ans: 
4-2 x 45 = 4(-2+5) = 43

Ques 6: (-3)a/(-3)2
Ans:  (-3)(a-2)

Ques 7: (3)2-3
Ans: 
(32)-3 = 3(2 x -3) = 3-6

Ques 8: 114/11x
Ans: 11(4-x)

Ques 9:  x2 x x3 x x5 
Ans: 
x2 x x3 x x= X (2 + 3 + 5) = x10

Ques 10: (52)x 
Ans: (52)x=5(2 * x)= 52x

Section - 2
Ques 11: 34 x 3x 3

Ans: 34 x 32 x 3
= 3(4 + 2 + 1) = 37

Ques 12: xx x6 /x2
Ans: x(5+6- 2) = x9

Ques 13:  56 x 54x / 54
Ans: 
5(6+ 4x—4) = 54x+2

Ques 14: y7 x y8 x y-6
Ans: yx y8 x y-6 
= y(7 + 8 + (-6)) 
= y9

Ques 15: x4/x-3
Ans: x(4-(-3)) = x7

Ques 16: z5 x z-3/z-8
Ans: z(5+(-3)-(-8))= z10

Ques 17: 32x x 36x/3-3y
Ans: 3(2x+6x-(-3y)) = 38x+3y

Ques 18: (x2)6 x x3
Ans: 
x(2 x 6 + 3) = x(12+3) = x15

Ques 19: (z6)x x z3x
Ans:
z(6 * x+3x) = z(6x+3x) = z9x

Ques 20: 53 x (54)y/(5y)3
Ans: 
5(3+(4xy)-(y x 3)) = 5(3+4y-3y)=  5y+3

Section - 3
Ques 21: Compute the sum.Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

 Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 22: Which of the following has the lowest value?
(A) (-3)4

(B) -33
(C) (-3)-3
(D) (-2)3
(E) 2-6
Ans: We are looking for the answer with the lowest value, so we can focus only on answers that are negative as these answers have lower values than any positive answers.
(A) (-3)4 will result in a positive number because 4 is an even power.
(B) -33 = -(3)3 = -27
The exponent is done before multiplication (by -1) because of the order of operations.
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT 
The value of this expression is positive.
-2 7 has the lowest value of the three answer choices that result in negative numbers. The correct answer is (B).

Ques 23: Compute the sum. Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
The first two terms in the expression are in fact the same. Because these terms are equal, when the second is subtracted from the first they cancel out leaving only the third term.

Ques 24: Which of the following is equal to (2/5)-3?
 Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT 
The correct answer is (E).
Note: when a problem asks you to find a different or more simplified version of the same thing, check your work against the answer choices frequently to ensure that you don’t simplify or manipulate too far.

Ques 25: Which of the following has a value less than 1 ? (Select all that apply)
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:
We are looking for values less than 1 so any expressions with negative values, zero itself, or values between 0 and 1 will work.
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Dividing a smaller positive number by a larger positive number will result in a number less than 1.
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Dividing a larger positive number by a smaller positive number will result in a number greater than 1.
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
This answer is negative; therefore, it is less than 1.
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Dividing a larger positive number by a smaller positive number will result in a number greater than 1.
(E) (-4)3 = -64
This answer is negative; therefore, it is less than 1.

Section - 4
Simplify the following expressions by finding common bases.

Ques 26: 83 x 26
Ans: 83 x 26 = (23)3 x 26 = 29 x 26 = 215

Ques 27: 492 x 77
Ans: (72)2 x 77 = 74 x 77 = 711

Ques 28: 254 x 1253
Ans: 
(52)4 x (53)3 = 58 x 59 = 517

Ques 29: 9-2 x 272
Ans: 
(32)-2 x (33)2 = 3-4 x 36 = 32 

Section - 5
Simplify the following expressions by pulling out as many common factors as possible.


Ques 31: 63 + 33 = (A) 35 (B) 39 (C) 2(33)
Ans: 
Begin by breaking 6 down into its prime factors.
63 + 33 =
(2 x 3)3 + 33 =
(23) (33) + 33
Now each term contains (33). Factor it out.
(23)(33) + 33 =
33(23 + 1) =
33(9) =
33(32) = 35
We have a match. The answer is A.

Ques 32: 813 + 274 = (A) 37(2) (B) 312(2) (C) 314
Ans: 
Both bases are powers of 3. Rewrite the bases and combine.
813 + 274 =
(34)+ (33)4 =
312 + 312 =
312(1 + 1) =
312(2)
We have a match. The answer is B.

Ques 33: 152- 52 = (A) 52(2) (B) 5223 (C) 5232
Ans:  
Begin by breaking 15 down into its prime factors.
152 - 52 =
(3 x 5)2 - 52 =
(32)(52) - 52
Now both terms contain 52. Factor it out.
(32)(52) - 52 =
52(32 - 1) =
52(9 - 1) =
52(8)
We still don’t have a match, but we can break 8 down into its prime factors.
52(8) =
52(23)
We have a match. The answer is B.

Section 6
Ques 34: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: 

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 35:Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: 
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 36: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 37:Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 38: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 39: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:
 

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 40: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 41: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT 

Ques 42: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: 
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Section - 7
Simplify the following roots. Not every answer will be an integer.
Ques 43: √32
Ans: 
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 44: √24
Ans: 
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 45: √180
Ans:
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 46: √490
Ans:
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 47: √450
Ans:
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 48: √135
Ans:
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 49: √224
Ans: 
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 50: √343
Ans: 
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Section - 8
Simplify the following roots. You will be able to completely eliminate the root in every question. Ex- press answers as integers.
Ques 51: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: Pull out the greatest common factor of 362 and 152, namely 32, to give
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT Both 32 and 169 are perfect squares (169 = 132), so
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 52: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: Pull out the greatest common factor of 352 and 212, namely 72, to give
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT Both 72 and 16 are perfect squares (16 = 42), so
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 53: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: Pull out the greatest common factor of 56 and 57, namely 56, to give
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT Both 62 and 5are perfect squares (56= 5x 53), so
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 54: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: Pull out the greatest common factor of 84 and 85, namely 84, to give
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Both 84 and 32 are perfect squares (84 = 82 x 82), so
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 55: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: Pull out the greatest common factor of 215, 213, and 212, namely 212, to give
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Both 212 and 32 are perfect squares

(212 = 26 x 26), Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 56: Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: Pull out the greatest common factor of 503 and 502, namely 502, to give
Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT
Both 502 and 72 are perfect squares, so Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT

The document Formulas & Solved Examples: Exponents & Roots Notes | Study Quantitative Aptitude for GMAT - GMAT is a part of the GMAT Course Quantitative Aptitude for GMAT.
All you need of GMAT at this link: GMAT
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