Roots and Exponents

Question 1

Un Squaring V/S Taking the square root.

Answer 1

Taking the square root of a number will always yield a positive result i.e √4 = 2

However, un squaring a variable will yield a positive and a negative result i.e-
If x^2 = 4 then x = +/-√4 = +/- 2 since both 2x2 and -2x-2 are equal to 4

Note-This principle applies to variables raised to any even power as even powers mask the sign of their base.

Question 2

To simplify a radical i.e √abcde

Answer 2

Prime Factorize the number and then pull out the largest perfect square/cube depending on which root is required i.e √1000 = √100 x10 = 10√10

Question 3

Approximating Square/Cube Roots. i.e Find √x where x is not a perfect square

Answer 3

We find the perfect squares / cubes before and after the number x and use them as boundaries. Then we approximate depending on which of the numbers x is closer to.
eg. √11 lies between √9=3 and √16=4 however since 9 is closer to 11 than 16 we estimate it to be 3.3 approximately.
Note- we use the same process for other roots.

Question 4

Rules for multiplying and dividing radicals i.e c√a x d√b and c√a / d√b, (2 properties)

Answer 4

1) The radicles must have the same index i.e they must both be squares or cubes etc.
2) The non-radical part of the term must only be multiplied with the non-radical part of the other term similarly the radical part of the term must only be multiplied with the radical part of the other term do not mix up the two

3) c√a x d√b = cd√ab i.e 2√5 x 3√5 = 6√25 = 6x5 =30
3) c√a / d√b = c√a / db i.e 4√125 / 2√5 = 2√125/5 = 2√25 = 2x5 = 10

Question 5

Addition or Subtraction of terms under a radical sing (√)

Answer 5

We must simplify the terms under the radical sign before we can take the square root i.e √a+b is not equal to √a+√b ie √16+25 = √41 = 4.1 approximately and √16+√25 = 4+5 = 9!

Question 6

Rules for addition and subtraction of radicals

i.e c√a + d√b (2 properties)

Answer 6

1) The radicles must have the same index i.e they must both be squares or cubes etc.
2) The radicle part of the term must be the same i.e we cannot add c√a + d√b unless a=d
eg- 4√3+5√3= 9√3 however we cannot add 3√5 and 4√3

Question 7

2 methods to simplify radicles in the denominator

Answer 7

A radical in the denominator is never considered to be simplified.

1) If there is only one radical term in the denominator we multiply both the numerator and denominator by that term to get rid of it i.e 2/√a = 2x√a/√ax√a =2√a/a in effect we just move the root up
2) Difference of squares- If the term in the denominator is a binomial then we multiply the numerator and denominator by its conjugate pair i.e 1 /a+√b = (1)(a-√b) /(a+√b)(a-√b) = a-√b/ a^2-b.
i. e we just multiply by the difference of squares to remove the √ from the denominator.

Question 8

Process for solving equations with square roots and caution!

Answer 8

1) Isolate the square root on one side of the equation
2) Square both sides of the equation and simplify (note the square of a number is always positive)
3) If we get more than one solution then plug those solutions back into the equation to make sure it satisfies equation else discard it.

Question 9

Caution when taking the square root of a binomial

Answer 9

When we take the square root of a binomial we must use the modulus sign as the binomial could be positive or negative in nature as even powers mask the base’s sign. I.e since √(x)^2 = IxI or +/- x then √(x+y)^2 = Ix+yI or +/- (x+y)

Question 10

The Base is greater than 1 and exponent is an even positive integer
i.e x > 1, n > 0 and n even

Answer 10

⇒ The result is larger.
I.e X^n > X
eg 4^2 = 16, 3^4 = 81

Question 11

The Base is greater than 1 and exponent is an odd positive integer greater than 1
i.e x > 1, n > 1 and n odd

Answer 11

⇒ The result is larger.
I.e X^n > X
eg 4^3 = 64, 2^5 = 32

Question 12

The Base is less than –1 and exponent is an even positive integer
I.e x < –1, n > 0 and n even

Answer 12

⇒ The result is larger.
I.e X^n > X
eg (-4)^2 = 16, (-3)^4 = 81

Question 13

The Base is less than –1 and exponent is an odd positive integer greater than 1
I.e x < –1, n > 1 and n odd

Answer 13

⇒ The result is smaller.
I.e X^n < X
eg (-4)^3 = -64, (-2)^5 = -32

Question 14

The Base is a positive proper fraction and exponent is an even positive integer
I.e 0 < x < 1, n > 0 and n even

Answer 14

⇒ The result is smaller.
I.e X^n < X
eg (1/4)^2 = 1/16, (1/3)^4 = 1/81

Question 15

The Base is a negative proper fraction and exponent is an even positive integer
i.e –1 < x < 0, n > 0 and n even

Answer 15

⇒ The result is larger.
I.e X^n > X
eg (- 1/4)^2 = 1/16, (-1/3)^4 = 1/81

Question 16

The Base is a positive proper fraction and exponent is an odd positive integer greater than 1
i.e 0 < x < 1, n > 1 and n odd

Answer 16

⇒ The result is smaller.
I.e X^n < X
eg (1/4)^3 = 1/64, (1/2)^5 = 1/32

Question 17

The Base is a negative proper fraction and exponent is an odd positive integer greater than 1
i.e –1 < x < 0, n > 1 and n odd

Answer 17

⇒ The result is larger.
I.e X^n > X
eg (- 1/4)^3 = -1/64, (-1/2)^5 = 1/32

Question 18

Zero raised to any positive power

i.e 0^n=

Answer 18

Is Zero
i.e 0^n= 0 where n > 0
eg 0^3=0 0^4=0

Question 19

One raised to any power

1^n=

Answer 19

Is One
i.e 1^n= 1
1^2=1, 1^3=1

Question 20

The Base is greater than 1 and exponent is a positive proper fraction
x > 1, 0 < n < 1

Answer 20

⇒ The result is smaller.
I.e X^n < X
eg. (4)^1/2 = Square Root 4 = 2
eg. (27)^1/3 = Cube Root 27 = 3

Question 21

The Base is a positive proper fraction and exponent is a positive proper fraction
0 < x < 1, 0 < n < 1

Answer 21

⇒ Result is larger
I.e X^n > X
eg. (1/4)^1/2 = Square Root 1/4 = 1/2
eg. (1/27)^1/3 = Cube Root 1/27 = 1/3

Question 22

If bases are equal then exponents…

If not then under what conditions?

Answer 22

Maybe Equal.
i.e if 3^5=3^x then x must be equal to 5.
However, this condition will not hold true if the bases are 1, 0 or -1. as these numbers raised to anything yield them yield a fixed value i.e 1^100 = 1^200 however 100 is not equal to 200

Question 23

If Bases are common then to multiply them…

Answer 23

Keep the base and the Exponents.
i.e X^a x X^b= X^a+b
5^5 x 5^4 = 5^5+4 = 5^9

Question 24

If Bases are common then to divide them…

Answer 24

Keep the base and subract Exponents.
i.e X^a / X^b= X^a-b
5^5 / 5^4 = 5^5-4 = 5^1 = 5

Question 25

Power to a power rule

Answer 25

When a base and its exponent are raised to a power then keep the base and multiply the exponent with the power that it is raised to.
I.e (X^a)^b= X^ab
(5^2)^3 = 5^23 = 5^6 = 15625

Question 26

How would we simplify an equation in which the bases are not the same?

Answer 26

Make them the same i.e Prime factorize all the bases and split the powers over the prime factors then group the common prime factors and use the rules of exponents to simplify.

Question 27

If the exponents of terms are common but the bases are different then to multiply them…

Answer 27

We keep the exponent common and multiply the bases.
i.e X^a x Y^a= (XY)^a
2^4 x 3^4= (23)^4 = 6^4 = 1296

Question 28

If the exponents of terms are common but the bases are different then to divide them…

Answer 28

We keep the exponent common and divide the bases.
i.e X^a / Y^a= (X/Y)^a
12^4 x 3^4= (12/3)^4 = 4^4 = 256

Question 29

Exponents can be distributed over…
Exponents can never be distributed over…
When distributed an exponent applies to…

Answer 29

An exponent can be distributed over multiplication or division never over addition or subtraction.

In the case of multiplication or division, the exponent applies to each term within the brackets.
i.e (4abc)^2 = 16(a^2)(b^2)(c^2)
(5/7) = (5)^2/(7)^2 = 25/79

Question 30

How are Radicles expressed in exponential form?

Answer 30

Radicals are expressed as the denominators of fractions in exponential form.
i.e A square root is the exponent 1/2 n cube roots are 1/3, Hence √x = x^1/2.

Question 31

How do we simplify a series of roots? ie each term is under multiple root signs

Answer 31

We split up each term and raise each term to its corresponding fractional exponent and then simplify.
i.e 3x√√3x√√√3 = 3^1/2 x 3^1/2x1/2 x 3^1/2x1/2x1/2
=3^1/2 x 3^1/4 x 3^1/8 = 3^1/2 + 1/4 + 1/8 = 3^7/8

Question 32

How do we remove the radicals across an equation?

Answer 32

We express the radicles in their exponential form and then take then raise the whole equation the LCM of the exponential denominators. Similar to removing fractions from equations.
i.e - 60th√X= 10th√2 => (X)^1/60 = (2)^1/10 => ((X)^1/60)^60 = ((2)^1/10)^60 => X = 2^6

Question 33

Method for comparing the magnitude of multiple Radicles

Answer 33

1) Express each radical as its fractional exponent form.
2) Raise each term the LCM of the exponential denominators of all the terms.
3) Cancel out all the denominators leaving only terms raised to a power making them easy to compare.
i.e is the 4th root of 4 greater than the 5th root of 7?
4th√4 = (4)^1/4 = ( (4)^1/4 )^20 = 4^5 = 1024
5th√7 = (7)^1/5 = ( (7)^1/5 )^20 = 7^4 = 2401
Therefore 5th√7 > 4th√4

Note- If the bases are raised to very large exponents we can use a similar process to factor out the Greatest Common Factor thus making the exponents smaller.

Question 34

A base raised to a negative exponent implies

Answer 34

That the term was in the denominator and it has been moved to the numerator i.e if we want to make the exponent positive again we must move it to the denominator again.
i.e X^-Y = 1/X^Y if X is not equal to zero
eg - 2^-2 = 1/2^2 = 1/4

Question 35

The process to simplify exponents expressed as quadratic equations? (4)

Answer 35

1) Foil the Quadratic equation
2) reduce each base to its prime factors and distribute the exponents.
3) Combine and cancel exponents according to their rules where ever possible.
4) We should be left with the same base on both sides of the equation in which case drop the base and simplify

Question 36

Any non zero bases raised to zero equals

Answer 36

To one i.e 2^0 = 1
However zero raised to zero is not defined and not tested on the GMAT however we cannot assume a variable is not equal to zero for a DS question.

Question 37

Any Base raised to one equals

Answer 37

To the base itself i.e 2^1 = 2

Question 38

Addition or Subtraction of like bases with like radicles we should attempt to…

Answer 38

Factor out the common base and radicles

i.e X^4 + X^4 + X^4 + X^4 = X^4 (1+1+1+1) = 4X^4

Question 39

The trick for adding Bases with equal exponents?

Answer 39

We can factor out the common base and exponent thus simplifying.
i.e 2^n + 2^n = 2^n(1+1) = 2^n(2) = 2^n(1+1)
3^4 + 3^4 + 3^4 = 3^4 (1+1+1) = 3^4(3) = 3^5

Question 40

Addition or Subtraction of Fractions that contain exponents in their denominators

Answer 40

Similar to addition or subtraction of normal fraction we multiply both terms by the LCM to equalize the denominators post which we can perform the requisite operation.

Question 41

Estimating the result of exponential terms that differ greatly in magnitude post addition or subtraction

Answer 41

The result will always be very close to the larger exponent as the smaller one will not make any significant difference to its size

Question 42

Powers of ten i.e 10^x and 10^-y mean how many digits.

Answer 42

10 raised to any positive power implies 1 with that many zeros to the right of it hence 10^x = 1+ “x” number of zeros to the right of 1. i.e 10^2 = 100 i.e 1 with 2 zeros to the right of one.
Similarly, 10^-y is 1/10^y i.e is 1 upon 1 with that many zeros to the right of it hence 10^-y is 1/10^y = 1/ 1+ “y” number of zeros to the right of 1. i.e 10^-2 = 1/10^2 = 1/100 i.e 1 with 2 zeros to the right of one which could also be expressed as 0.01 i.e for 10^-y we move the decimal place by y places i.e 10^-2 = 0.01 i.e we move the decimal by 2 places.

Question 43

When do we use Scientific Notation and how would you express a number in Scientific Notation?

Answer 43

Scientific Notation is used to express extremely large and extremely small numbers i.e those close to 1.
A number “n” = a x 10^b when expressed in scientific notation where -
a is a coefficient between 1 and 10 including one but excluding 10 and
b is the power of 10 for very small numbers this is -ve.
i.e 3.14= 3.14 x 10^0,
93000000 = 9.3 x 10^7 note 7 is the number of places we move the decimal to the left
0.0000678 = 6.78 x 10^-5 note -5 is the number of places we move the decimal to the right.

Question 44

Multiplication or division with scientific notation

Answer 44

A number “n” = a x 10^b when expressed in scientific notation where -
a is a coefficient between 1 and 10 including one but excluding 10 hence we multiply and divide the coefficients of the two numbers and simplify the exponential value of 10 by either adding or subtracting powers depending on multiplication or division according to the rules of exponents.

I.e (4.8 x 10^8)/(0.4 x 10^3) = (4.8/0.4) (10^8-3) = (48/4) (10^5) = 12 x 10^5 = 1.2 x 10^6

Question 45

Square roots of extremely large numbers ending in zeros and extremely small numbers with many decimal places.

Answer 45

For a number to be a perfect square it will end in an even number of zeros. The square root of such a number will have exactly half the number of trailing zeros. i.e √100 = 10^2x1/2 = 10.

A decimal is a perfect square if both the decimal and its square root have a finite number of digits and the number of decimal places it has is even. In which case the square root will have exactly half the number of decimal places that the number has.
i.e √0.0004 = √4 x √10^-4 = 2 x 10^-4x1/2 = 2x10^-2 = 0.02

Question 46

Cube roots of extremely large numbers ending in zeros and extremely small numbers with many decimal places.

Answer 46

For a number to be a perfect Cube it will end in a number of zeros which is divisible by 3. The Cube root of such a number will have exactly 1/3rd the number of trailing zeros. i.e √1000000 = 10^6x1/3 = 10^2 = 100

A decimal is a perfect Cube if both the decimal and its cube root have a finite number of digits and the number of decimal places it has is divisible by 3. In which case the Cube root will have exactly 1/3rd the number of decimal places that the number has.
i.e (0.000027)^1/3 = (27)^1/3 x 10^-6x1/3 = 3 x 10^-2 = 0.03

Question 47

The trick for squaring decimals with a large number of zeros?

Answer 47

Multiply the non zero part by itself and then count the number of decimals in the original number and double it.
I.e (0.000005)^2 =0.000000000025 => 5^2= 25 and the orignal number had 6 decimal places i.e its square will have 6x2= 12 decimal places since-
(0.000005)^2 = (5)^2 x 10^-6x2 = 25 x 10^-12