Q4) Roots and exponents

Question 1

Must know: Square root of:

A) a number e.g., 4
B) a variable e.g., x
c) a variable squared e.g., x^2
d) a number squared e.g., (-4)^2

Answer 1

Square root of a non-negative number: sqrt(#) = # (always positive or 0)
> This is because the radicand must always be NON-NEGATIVE

Square root of a VARIABLE: sqrt(x) —-> means always x >= 0
> This is because the radicand must always be NON-NEGATIVE

Square root of a VARIABLE SQUARED:
sqrt(x^2) = | x |

> replacing the absolute value signs with +/- —> sqrt(x^2) = +/- x
same thing applies to square root of a BINOMIAL SQUARED

e.g. sqrt((x+y)^2) = |x+y|

Evaluate the CASES for the value of what’s inside the absolute value
e.g., case 1 is x+y > 0 —> |x+y| = x+y and case 2 is x+y < 0 —> |x+y| = -(x+y)

Square root of a number squared:
sqrt(#^2) = | # |

Question 2

Even and odd indexed roots

Answer 2

Even indexed roots –>
> answer is always NON-NEGATIVE
> radicand (inside root) is ALSO always NON-NEGATIVE

e.g., sqrt(x) can only produce a real number if x >=0

Algebraically: (x^n)^(1/n) = | x |

> if x > 0, | x | = x
if x < 0, | x | = -x

Odd indexed roots –> watch out for + or - signs
> radicand and answer can be + or - or 0

Algebraically: (x^n)^(1/n) = x

Question 3

If n is positive, is n an integer?

(1) sqrt(n) =/ integer
(2) sqrt(n) < 1

Answer 3

Concept: perfect squares and square roots
> recall that the square root of a perfect square produces an INTEGER

(1) sqrt(n) =/ integer
n =/ integer^2

n is NOT a perfect square, but does not tell us whether n is an integer or not

e.g., sqrt(2) is not an integer and sqrt(4.1) is not an integer

NS

(2) sqrt(n) < 1
n < 1 (and since n > 0, n must be a fraction)

S (answer is always no)

Testing also proves this:

Sqrt(1/4) = 1/2 –> n is not an integer
Sqrt(1) = 1 (invalid example)
Sqrt(2) = 1.41 (greater than 1, invalid example)

Learning: You CAN take the square root of FRACTIONS / decimals

Question 4

Perfect cubes and cube root

Answer 4

> Perfect cubes have exponents in prime factorization form that are MULTIPLES OF 3
Cube root of a perfect cube produce an integer (+, -, or 0)
radicand can be +, - or 0

(x)^1/3 = integer
or
(-x)^1/3 = integer

Question 5

Memorize: approximation of square roots of non-perfect single digit numbers

Sqrt(2)
Sqrt(3)
Sqrt(5)
Sqrt(6)
Sqrt(7)
Sqrt(8)

Answer 5

Sqrt(2) = 1.4 (valentine’s day)
Sqrt(3) = 1.7 (st. patrick’s day)
Sqrt(5) = 2.2 (go up by 0.2)
Sqrt(6) = 2.4
Sqrt(7) = 2.6
Sqrt(8) = 2.8

Question 6

How do you approximate larger non-perfect square roots?

e.g., which of the following square root expressions is the largest?

e.g., what is the approximate value of sqrt(70)?

e.g., the expression sqrt(12 + 2sqrt(11)) - sqrt(12 - 2sqrt(11)) is equal to which of the following?

Answer 6

Create a RANGE using your knowledge of perfect squares immediately smaller and immediately larger than the radicand

e.g., 70 is between 64 and 81

Sqrt(70) is between sqrt(64) and sqrt(81)

You might see applications of approximation of NON PERFECT roots
e.g., the expression sqrt(12 + 2sqrt(11)) - sqrt(12 - 2sqrt(11)) is equal to which of the following?

Question 7

Addition/subtraction and multiplication/division of radicals

Answer 7

Multiplication/division –> can combine radicands under the radical ONLY IF the index are the same

e.g., sqrt(4)sqrt(5) = sqrt(45)
e.g., sqrt(80/10) = sqrt(80)/sqrt(10)

Addition/subtraction –> CANNOT combine radicands under the radical

e.g., sqrt(a) - sqrt(b) =/ sqrt(a - b)

BUT you can treat radicals with the same root index AND same radicand as LIKE TERMS during simplification
> combine coefficients

e.g., 5sqrt(4) + 4sqrt(4) = 9*sqrt(4)

Question 8

Can you keep radicals in the denominator of a fraction in GMAT?

Answer 8

No!
> simplified form must have all radicals REMOVED from the denominator of a fraction via two methods

1) Rationalizing the denominator (when there is a single radical in the denominator)
> multiply top and bottom of fraction by the radicand

2) Rationalizing the denominator (when there is a BINOMIAL containing a radical)
> Multiply top and bottom by the CONJUGATE of the denominator (creates difference of squares)

e.g., a-sqrt(b) has a conjugate of a+sqrt(b)

Question 9

What should you get into the habit doing every time you solve an equation that involves a SQUARE ROOT or ABSOLUTE VALUE?

Answer 9

Plug back your answer into the original equation and CHECK that it works ** (eliminate invalid answers)

> especially important if you end up with more than one solution for x
also important to note that if an absolute value of an expression is equal to a negative number, there will be NO solutions to that equation

Question 10

What are special numbers to test as bases of exponent questions?

Answer 10

1
-1
0

1^anything = 1
-1^even = 1
-1^odd = -1
0^anything = 0
(most crucial for UNKNOWN BASES raised to exponents)

Therefore, if a=/ -1, 1, or 0 and a^x = a^y (same base), then x = y
> you can “drop the bases” and solve the equation involving the exponents

Question 11

Exponent rules - when do you add/subtract exponents vs multiply exponents?

Answer 11

ADD / Subtract exponents only when you are dealing with
> Multiplication of like bases (for addition) or division of like bases (for subtraction)

Multiply exponents if you have a POWER raised to a POWER
> sprinkle the exponent to inner terms that are multiplied together

e.g., (ab)^2 = a^2*b^2

Remember that the reverse operation is also true –> multiplication of different bases and LIKE exponents, keep the common exponent and multiply the bases

(a^2)*(b^2) = (ab)^2

Same thing applies to division of different bases and LIKE exponents –> keep the common exponent and divide the bases

Question 12

When you see large bases (composite number) with exponents, what should you think of doing?

Answer 12

Prime factorize the bases (but remember to keep the parentheses around the prime factorization

> applies also to equations (helps you solve the unknowns in the exponents)

Question 13

Dealing with nested roots

e.g., sqrt(3 * sqrt(3 *sqrt(3)))

Answer 13

> start with the inner most root
convert all roots into exponent form
you end up multiplying series of roots together

OR short cut:
> figure out how many roots each number is under = SAME NUMBER of exponents multiplied together
> Start with FAR LEFT

e.g., first 3 is under only one root sign –> sqrt(3); second 3 is under two root signs —> 3^(1/3)^2; third 3 is under three root signs –> 3^(1/3)^3

Question 14

*** How do you compare radicals and exponents and understand their relative sizes?

e.g., which is larger 4^(1/4) or 7^(1/5)?

e.g., which is larger, 5^50 or 7^25?

Answer 14

**CONCEPT: If x, y and m are POSITIVE, then x > y ONLY IF x^m > y^m
(want to make the EXPONENTS the same if you cannot make the bases the same)

> x and y represent RADICALS or BASES

> When dealing with RADICALS –> Start by expressing each radical as an EXPONENT. Then raise all expressions to the least common denominator of the exponents and EVALUATE the power

e.g., raise both 4^(1/4) and 7^(1/5) by LCD of 20

4^5 < 7^4

therefore 4^(1/4) < 7^(1/5)

> When dealing with WHOLE NUMBER BASES –> Raise all expressions to the RECIPROCAL of the GCF in order to “scale down” the numbers

e.g., raise both 5^50 and 7^25 by 1/GCF (1/25)

5^2 > 7

therefore 5^50 > 7^25

Why this works?
> scaling up or down ALL numbers

Question 15

1/4 * (40^100) is equal to which of the following?

(A) 4^99
(B) 20^180
(C) 420^180
(D) 400(40^98)
(E) 10*(40^98)

Answer 15

Ans D

Concept: Be on the lookout for different ways to FACTOR VALUES written in EXPONENTIAL notation (especially when dealing with very large exponents)

40^100 = 40^99 * 40 = 40^98 * 40^2 = 40^97 * 40^3 …. etc.

> you can SPLIT up exponents sharing the same base
you can COMBINE exponents sharing the same power

Question 16

Define “binomial”

Answer 16

Sum or difference of TWO terms involving unknowns

e.g., x + y, 3 - a

Question 17

Squaring binomials

Answer 17

Squaring binomial refers to multiplying the binomial BY ITSELF and then using the FOIL process to expand the product of binomials

(a+b)^2 = (a+b)*(a+b) = a^2 + 2ab = b^2

Question 18

Fractional base raised to a negative exponent rule of thumb

Answer 18

if x=/ 0 and y=/0, then (x/y)^-z = (y/x)^z

> flip the fraction AND make exponent positive

Question 19

Quadratic expressions as exponents

e.g., 2^(a+b)^2 = 2^(a-b)^2

Answer 19

Follow the same rules about quadratic equations (e.g., foiling) and roots and exponents (like bases etc.)

Question 20

** Short cut when ADDING like bases with equal exponents

e.g., 2^3 + 2^3
e.g., 2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6
e.g., 25 + 45^2 + 45^3 …

Answer 20

If you are ADDING the SAME NUMBER of terms as the base together with equal exponents –> the result is the same base raised to ONE HIGHER exponent

2^n + 2^n = 2^(n+1)

3^n + 3^n + 3^n = 3^(n+1)

4^n + 4^n + 4^n + 4^n = 4^(n+1)

APPLICATION OF THIS TYPE OF QUESTION involves LIKE BASES with INCREASING EXPONENTS
> IF the pattern is not quite noticeable to you at first (same bases, raised to increasing exponents) –> TACKLE PAIRS of terms in increments

e.g., First tackle 25 and 4^5^2 = 5^2(1+4) = 5^2*5 = 5^3

Question 21

Must know rule about the relationship between x, x^2 and sqrt(x) if you know that 0 < x < 1 …

Answer 21

x^2 < x < sqrt(x)

e.g., x = 1/4
x^2 = 1/16
sqrt(x) = 1/2

** In general:
> if you take the ROOT of a fraction, the result is LARGER
> otherwise, the result depends on whether the (1) base is positive/negative (2) exponent is even or odd (3) base is fraction or whole number (4) exponent is fraction or whole number (5) exponent is positive or negative

> when in doubt, use an easy test number for PS

Question 22

What can the value of x be?

x^2 > x

Answer 22

(1) x can be negative fraction or integer

(2) x can be positive integer

THINK OF A NUMBER line for every number that you can test for the base (5 possible bases)

Question 23

Number line test values when determining the value of a base number after it has been changed (raised by an exponent or square root taken)

Answer 23

NUMBER LINE, from left to right:

5 possible BASES…
(1) x < -1
(2) -1 < x < 0
(3) x = 0
(4) 0 < x < 1
(5) x > 1

Question 24

What can the value of x be?

x^3 > x

Answer 24

(1) Negative fraction
(2) Positive integer

Question 25

What can the value of x be?

x^3 < x

Answer 25

(1) Negative integer
(2) Positive fraction (0 < x < 1)

Question 26

What can the value of x be?

Sqrt(x) < x

Answer 26

(1) Positive integer

Question 27

What can the value of x be?

Sqrt(x) > x

Answer 27

(1) Positive fraction (0 < x < 1)

Question 28

What can the value of x be?

x^2 < x

Answer 28

(1) Positive fraction

Question 29

Estimating with exponents

e.g., what is the approximate value of 9^8 - 2^12. Is it closer to 9^8 or 9^7?

e.g., the product of 1/sqrt(256) * 1/sqrt(64) * 1/100 * 1/10^3 is closest to which of the following?

e.g., (27^5 + 9^3)/(3^12 + 81^2) is closest to which of the following?

Answer 29

Key concept:
> two numbers with the SAME BASE and exponents that differ by as little as 1 can be VASTLY DIFFERENT from each other
(the number with the higher exponent is MUCH LARGER)
> also, the difference is more pronounced when BASES AND EXPONENTS are relatively large

In the example, we know that the value being subtracted (2^12) is very small compared to 9^8. So 9^8 - 2^12 is closer to 9^8

IN GENERAL strategy for solving:
> Simplify as much as possible
> approximate terms (incl. numerator and denominator separately)
e.g., 3^9 + 1 = ~3^9

> You can also convert numbers into the NEAREST POWER OF 10 (including positive and negative exponents) to make it easier to estimate the PRODUCT of fractions or powers

e.g., 1/16 * 1/8 = 1/128 => close to 1/100

Question 30

Square roots of large and small perfect squares

e.g., What is the square root of 81,000,000?

e.g., what is the square root of 0.000081?

Answer 30

A) When a perfect square is an integer with an EVEN NUMBER of zeros in the end => square root of such a perfect square will have exactly HALF THE NUMBER OF ZEROS to the right of the final nonzero digit
> This is because the power of 10 has an EVEN exponent that gets DIVIDED BY 2 when the square root gets taken
e.g., sqrt(81,000,000) –> 6 zeros, ans = 9000

B) When a perfect square is a DECIMAL => its square root will have exactly HALF the number of DECIMAL PLACES to the right of the decimal
> Yes, perfect squares can be decimals
> A perfect square decimal must have an EVEN NUMBER of DECIMAL places (not zeros) to the right of the decimal
> AND square root must have a finite number of decimal places
e.g., sqrt(0.000081) = sqrt(81/10^6) = 9/10^3 = 0.009
> can reason it out too, 0.000081 has 6 decimal places, so its square root will have 3 decimal places. sqrt(81) = 9, so there must be 2 zeros after the decimal point. Ans = 0.009

Question 31

Cube roots of large and small perfect squares

e.g., What is the cube root of 27,000,000?

e.g., what is the cube root of 0.000027?

Answer 31

A) When a perfect cube is an integer with zeros in the end => cube root of such a perfect cube will have exactly 1/3 THE NUMBER OF ZEROS to the right of the final nonzero digit
> This is because the power of 10 has an exponent that gets DIVIDED BY 3 when the square root gets taken
e.g., Cube root of (27,000,000) –> 6 zeros, ans = 100

B When a perfect cube is a decimal => cube root of such a perfect cube will have exactly 1/3 THE NUMBER OF DECIMAL PLACES as the original perfect cube
e.g., cube root of 0.000027 –> 6 decimal places, ans 0.03

Question 32

** Squaring / Cubing decimals with zeros

e.g., (0.000005)^2 = ?
e.g., (0.03)^3

Answer 32

After squaring a decimal, the NUMBER OF DECIMAL PLACES to the right of the decimal is DOUBLED
e.g., 2 decimal places => 4 decimal places

After cubing a decimal, the NUMBER OF DECIMAL PLACES to the right of the decimal is TRIPLED
e.g., 2 decimal places => 6 decimal places

Strategy:
A) Reason it out
> 0.000005 has 6 decimal places, 0.000005^2 will have 2 * 6 = 12 decimal places. Since 25 already has 2 decimal places, we need to add 10 more zeros in front to yield the full 12 decimals.
> similarly, 0.03 has 2 decimal places, 0.03^3 will have 6 decimal places. DON’T FORGET that 3^3 = 27, so we need to have 4 more zeros in front to yield the full 6 decimals = 0.000027

B) Can convert the decimal to a fraction (easier if denom is power of 10) and determine the number of decimal places and zeros

(0.000005)^2 = (5/10^6)^2
= 25/10^12
= 0.25 * 10^-10 —> 10 zeros plus two digits
= 0.000000000025

(0.03)^3 = (3/100)^3
= 27/10^6
= 0.27 * 10^-4 —–> 4 zeros plus 2 and 7
= 0.000027

Question 33

Exponents with improper fractions

e.g., x^13/12

Answer 33

You can SPLIT the exponent so that it becomes a MIXED fraction

e.g., x^13/12 = x^(1 + 1/2) = x * sqrt(x)

Question 34

Comparing fractions with exponents

e.g., which of the following fractions is the largest

(A) 1/(2^2 *3)
(B) 2/(3^2)

etc.

Answer 34

Same approaches to comparing any other fraction, though it is EASIEST to convert all options to the SAME DENOMINATOR = Least Common Denominator = highest powers of prime * unique primes

Question 35

**Nested roots

Ex #1 –> What is sqrt(5 * sqrt(5 * sqrt(5 * sqrt(5))))

Ex #2 –> What is sqrt(sqrt(sqrt(5)))

Answer 35

Pay attention to:
> Number of nested roots = exponents
> ** AND numbers and operations

Strategy:
> Start with inner most radicand and convert from sign to exponent
> Keep terms as a PRODUCT e.g., 5 * 5^1/2
> Make your way outside by “unravelling” the sqrt and applying the exponent to each new radicand
> Avoid adding exponents (simplifying common bases) until the very end

Ex #1: don’t forget the 5 * __ (not just 5 to the power of 1/16)

Ans: 5^(15/16)

Question 36

Tip when dealing with PS involving fractional bases raised to powers

e.g., [1/(2^2)]^(2/3)

Answer 36

If the NUMERATOR equals 1, you can easily pull out the exponent so there is NO exponent in the denominator

e.g., [1/(2^2)]^(2/3)

[(1/2)^2]^(2/3)

Remember:
a^2/b^2 = (a/b)^2

Question 37

Multiple exponents

e.g., 3^(a-b)^(a-b)

e.g., (((5^(1/2))^(1/2))^(1/2))

Answer 37

Just MULTIPLY the exponents together, keeping the base the same

Question 38

What happens when you are asked to solve an equation involving powers but the BASES are not the same

e.g., (8/27)^-x * (6/4)^(2x-4) = 1

Answer 38

(1) Manipulate the bases to produce one common base (“like bases”)
(2) Another tip, when you see fractions raised to negative exponents, get rid of the negative exponent by FLIPPING the fraction

Question 39

Exponent PS - situations where you see a common exponent term being repeated over and over again

e.g., (2^8x + 2^8x) * (3^10y + 3^10y + 3^10y)

Answer 39

(1) Factor out common terms
(2) DO NOT blindly combine coefficients
(3) Instead, remember when you have LIKE BASES and multiplication, ADD the exponents
(same thing with division = subtract the exponents)
(4) Your goal is to match BASES on both sides of the equation

e.g., (2^8x + 2^8x) * (3^10y + 3^10y + 3^10y)

= 2^8x(2)(3^10y)(3) —–> DO NOT combine 23
= 2^(8x+1) * 3^(10y+1)

Question 40

Solving equations and expressions involving square roots

e.g., the expression sqrt(12 + 2sqrt(11)) - sqrt(12 - 2sqrt(11)) is equal to which of the following?

Answer 40

Note: even if an expression does not explicitly present itself as an equation, you can set the expression = x and then remove square roots by SQUARING both sides!!

Question 41

What should you be careful of when you see BINOMIALS squared and asked about solutions?

e.g., DS: Is n = 2m +3?
(1) (n-3)^2 = 4m^2
(2) n^2 = 4m^2 + 12m + 9

Answer 41

> BEFORE you can take the square root, make sure to EXPRESS AS A BINOMIAL
e.g., 4m^2 + 12m + 9 = (2m + 3)^2

> Then, whenever you take the SQUARE ROOT of an unknown or binomial expression, remember there can be TWO SOLUTIONS

sqrt[(x+y)^2] = | x + y |
sqrt(4m^2) = 2*|m|

(just need to check when they are both equal to each other and when they are opposites)

So might not be sufficient alone

Question 42

If you see a number or variable raised to an UNNOWN exponent that EQUALS 1, what are the possible scenarios (bases and exponents)?

e.g., (x+y)^n = 1

Answer 42

(1) exponent could equal 0 (and base can be ANYTHING)

(2) Base could equal 1 (and exponent could be ANYTHING)

(3) Base could equal -1 and exponent could be EVEN