Formulas & Solved Examples: Exponents & Roots | Mathematics & Pedagogy Paper 2 for CTET & TET Exams - CTET & State TET PDF Download

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 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: 

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

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

7. Operations involving the same bases: 

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

8. Fraction as power: 

Roots

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

Formulas of Roots 

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 7⁹ 
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 4⁵ 
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: 11⁴/11^x
Ans: 11^(4-x)

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

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

Section - 2
Ques 11: 3⁴ x 3² x 3
Ans: 3⁴ x 3² x 3
= 3^{(4 + 2 + 1)} = 37

Ques 12: x⁵ x x⁶ /x²
Ans: x^{(5+6- 2)} = x⁹

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

Ques 14: y⁷ x y⁸ x y-⁶
Ans: y⁷ x 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: 3^2x x 3^6x/3^-3y
Ans: 3^{(2x+6x-(-3y))} = 3^8x+3y

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

Ques 19: (z⁶)^x x z^3x
Ans: z(6 * x+3x) = z(6x+3x) = z^9x

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

Section - 3
Ques 21: Compute the sum.
Ans:
 

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.

 
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.
Ans:

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?
 

Ans:
 
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)


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.

Dividing a smaller positive number by a larger positive number will result in a number less than 1.

Dividing a larger positive number by a smaller positive number will result in a number greater than 1.

This answer is negative; therefore, it is less than 1.

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)³ + (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:
Ans: 


Ques 35:
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Ques 36:
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Ques 37:
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Ques 38:
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Ques 39:
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Ques 40:
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Ques 41:
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Ques 42:
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Section - 7
Simplify the following roots. Not every answer will be an integer.
Ques 43: √32
Ans: 


Ques 44: √24
Ans: 


Ques 45: √180
Ans:


Ques 46: √490
Ans:


Ques 47: √450
Ans:


Ques 48: √135
Ans:


Ques 49: √224
Ans: 


Ques 50: √343
Ans: 


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:
Ans: Pull out the greatest common factor of 36² and 15², namely 3², to give
 Both 3² and 169 are perfect squares (169 = 13²), so


Ques 52:
Ans: Pull out the greatest common factor of 35² and 21², namely 7², to give
 Both 7² and 16 are perfect squares (16 = 4²), so


Ques 53:
Ans: Pull out the greatest common factor of 5⁶ and 5⁷, namely 5⁶, to give
 Both 6² and 5⁶ are perfect squares (5⁶= 5³ x 5³), so


Ques 54:
Ans: Pull out the greatest common factor of 8⁴ and 8⁵, namely 8⁴, to give

Both 8⁴ and 3² are perfect squares (8⁴ = 8² x 8²), so


Ques 55:
Ans: Pull out the greatest common factor of 2¹⁵, 2¹³, and 2¹², namely 2¹², to give

Both 2¹² and 3² are perfect squares

(2¹² = 2⁶ x 2⁶),

Ques 56:
Ans: Pull out the greatest common factor of 50³ and 50², namely 50², to give

Both 50² and 7² are perfect squares, so