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CTET & State TET Exam  >  CTET & State TET Notes  >  Mathematics & Pedagogy Paper 2 for CTET & TET Exams  >  Formulas & Solved Examples: Exponents & Roots

Formulas & Solved Examples Exponents & Roots - & Pedagogy Paper 2 for CTET

Exponents

Exponents are a concise way to show that a number (the base) is multiplied by itself several times.

Exponents
  • Notation: an means a is multiplied by itself n times; a is the base and n is the exponent.
  • Interpretation: the exponent tells how many copies of the base are multiplied together.

Formulas of Exponents

  • 1. Exponents one and zero:

    a0 = 1 for any nonzero a.

    a1 = a.

    Note: the expression 00 is indeterminate in many contexts; the GMAT does not test 00.

  • 2. Powers of zero:

    If n > 0 then 0n = 0.

    If n < 0 then 0n is undefined (division by zero would be implied).

  • 3. Powers of one:

    1n = 1 for any integer n.

  • 4. Negative powers:

    a-n = 1 / an for a ≠ 0.

    Important: 0 cannot be raised to a negative power because that requires division by zero; for example 0-1 is undefined.

    Formulas of Exponents
  • 5. Powers of -1:

    (-1)n = 1 if n is even, and (-1)n = -1 if n is odd.

  • 6. Operations with same exponents (different bases):

    If two factors have the same exponent, multiply or divide the bases and keep the common exponent:

    (ab)n = an bn

    Formulas of Exponents
  • 7. Operations with same base:

    When multiplying with the same base, keep the base and add exponents: am · an = am+n.

    When dividing with the same base, keep the base and subtract exponents: am / an = am-n.

    Formulas of Exponents
  • 8. Power of a power and fractional powers:

    (am)n = am·n.

    ap/q = q√(ap) = (q√a)p where q > 0 (principal qth root).

    Formulas of Exponents
MULTIPLE CHOICE QUESTION

Try yourself: The value of 2-2 is:

A

4

B

1/4

C

2

D

1/2

Roots

Roots (radicals) reverse the action of exponents. For example, if x2 = 16 then √16 = 4 (principal square root).

Roots

Formulas of Roots

  • Relation to exponents:

    √[q]{a} = a1/q. In particular, √a = a1/2.

  • Properties:

    q√(ab) = q√a · q√b for non-negative a and b when q is even; the property extends for principal roots where defined.

    q√(am) = (q√a)m = am/q.

    Formulas of Roots
MULTIPLE CHOICE QUESTION

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

A

224

B

 2-5

C

29

D

2-9

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 (for example, write 23 rather than 8).

Ques 1. x5 x x3
Ans:

x5 x x3 =

x5+3 =

x8

Ques 2: 76 x 79
Ans:

76 x 79 =

76+9 =

715

Ques 3: 55/55
Ans:

55 / 55 =

55-5 =

50 =

1

Ques 4: (a3)2
Ans:

(a3)2 =

a3·2 =

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 / (-3)2 =

(-3)a-2

Ques 7: (3)2-3
Ans:

32-3 =

3-1

Ques 8: 114/11x
Ans:

114 / 11x =

114-x

Ques 9: x2 x x3 x x5
Ans:

x2 x x3 x x5 =

x2+3+5 =

x10

Ques 10: (52)x
Ans:

(52)x =

52x

Section - 2

Ques 11: 34 x 32 x 3
Ans:

34 x 32 x 3 =

34+2+1 =

37

Ques 12: x5 x x6 / x2
Ans:

x5 x x6 / x2 =

x5+6-2 =

x9

Ques 13: 56 x 54x / 54
Ans:

56 x 54x / 54 =

56+4x-4 =

54x+2

Ques 14: y7 x y8 x y-6
Ans:

y7 x y8 x y-6 =

y7+8+(-6) =

y9

Ques 15: x4/x-3
Ans:

x4 / x-3 =

x4-(-3) =

x7

Ques 16: z5 x z-3 / z-8
Ans:

z5 x z-3 / z-8 =

z5+(-3)-(-8) =

z10

Ques 17: 32x x 36x / 3-3y
Ans:

32x x 36x / 3-3y =

32x+6x-(-3y) =

38x+3y

Ques 18: (x2)6 x x3
Ans:

(x2)6 x x3 =

x2·6 x x3 =

x12+3 =

x15

Ques 19: (z6)x x z3x
Ans:

(z6)x x z3x =

z6x x z3x =

z6x+3x =

z9x

Ques 20: 53 x (54)y / (5y)3
Ans:

(54)y = 54y and (5y)3 = 53y.

Therefore, 53 x 54y / 53y =

53+4y-3y =

5y+3

Section - 3

Ques 21: Compute the sum.

Section - 3

Ans:

Section - 3
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 need the smallest (lowest) numerical value. Positive numbers are greater than any negative number, so consider only negative options.

(A) (-3)4 is positive because the exponent is even.

(B) -33 = -(33) = -27.

(C) (-3)-3 = 1 / (-3)3 = -1/27 (negative but ~ -0.037).

(D) (-2)3 = -8.

(E) 2-6 = 1 / 64 (positive).

Compare the negative values: -27, -8, and -1/27. The smallest is -27. Thus the correct answer is (B).

Section - 3
Section - 3

Ques 23: Compute the sum.

Section - 3

Ans:

Section - 3

The first two terms are equal and cancel when one is subtracted from the other, leaving only the third term.

Ques 24: Which of the following is equal to (2/5)-3?

Section - 3
Section - 3

Ans:

Section - 3

The correct answer is (E).

Note: when simplifying, check choices to avoid over-manipulation.

Ques 25: Which of the following has a value less than 1? (Select all that apply)

Section - 3
Section - 3

Ans:

Section - 3

Dividing a smaller positive number by a larger positive number gives a value between 0 and 1.

Section - 3

Dividing a larger positive number by a smaller positive number gives a value greater than 1.

Section - 3

This expression is negative, so it is less than 1.

Section - 3

Dividing a larger positive number by a smaller positive number gives a value greater than 1.

(E) (-4)3 = -64, which is negative and therefore less than 1.

Section - 4

Simplify the following expressions by finding common bases.

Ques 26: 83 x 26
Ans:

8 = 23, so 83 = (23)3 = 29.

Then 29 x 26 = 215.

Ques 27: 492 x 77
Ans:

49 = 72, so 492 = (72)2 = 74.

74 x 77 = 711.

Ques 28: 254 x 1253
Ans:

25 = 52 so 254 = 58.

125 = 53 so 1253 = 59.

58 x 59 = 517.

Ques 29: 9-2 x 272
Ans:

9 = 32, so 9-2 = (32)-2 = 3-4.

27 = 33, so 272 = 36.

3-4 x 36 = 32.

Section - 5

Simplify the following expressions by pulling out common factors.

Ques 31: 63 + 33 = (A) 35 (B) 39 (C) 2(33)
Ans:

6 = 2 x 3, so 63 = (2 x 3)3 = 23 x 33.

63 + 33 = 23 x 33 + 33.

Factor 33: 33(23 + 1) = 33(8 + 1) = 33 x 9 = 33 x 32 = 35.

Answer: A.

Ques 32: 813 + 274 = (A) 37(2) (B) 312(2) (C) 314
Ans:

81 = 34, so 813 = (34)3 = 312.

27 = 33, so 274 = (33)4 = 312.

312 + 312 = 312(1 + 1) = 2 · 312.

Answer: B.

Ques 33: 152 - 52 = (A) 52(2) (B) 5223 (C) 5232
Ans:

15 = 3 x 5, so 152 = (3 x 5)2 = 32 x 52.

152 - 52 = 32 x 52 - 52 = 52(32 - 1) = 52(9 - 1) = 52 x 8 = 52 x 23.

Answer: B.

Section 6

Ques 34:

Section 6

Ans:

Section 6

Ques 35:

Section 6

Ans:

Section 6

Ques 36:

Section 6

Ans:

Section 6

Ques 37:

Section 6

Ans:

Section 6

Ques 38:

Section 6

Ans:

Section 6

Ques 39:

Section 6

Ans:

Section 6

Ques 40:

Section 6

Ans:

Section 6

Ques 41:

Section 6

Ans:

Section 6

Ques 42:

Section 6

Ans:

Section 6

Section - 7

Simplify the following roots. Not every answer will be an integer.

Ques 43: √32
Ans:

Section - 7

Ques 44: √24
Ans:

Section - 7

Ques 45: √180
Ans:

Section - 7

Ques 46: √490
Ans:

Section - 7

Ques 47: √450
Ans:

Section - 7

Ques 48: √135
Ans:

Section - 7

Ques 49: √224
Ans:

Section - 7

Ques 50: √343
Ans:

Section - 7

Section - 8

Simplify the following roots. You will be able to completely eliminate the root in every question. Express answers as integers.

Ques 51:

Section - 8

Ans:

Pull out the greatest common perfect square factor of the terms under the square root.

Section - 8

Both 32 and 169 are perfect squares (169 = 132), so

Section - 8

Ques 52:

Section - 8

Ans:

Pull out the greatest common perfect square factor of the terms under the square root.

Section - 8

Both 72 and 16 are perfect squares (16 = 42), so

Section - 8

Ques 53:

Section - 8

Ans:

Pull out the greatest common perfect square factor of the terms under the square root, namely 56.

Section - 8

56 and the remaining factor are perfect squares, so

Section - 8

Ques 54:

Section - 8

Ans:

Pull out the greatest common perfect square factor 84.

Section - 8

Both 84 and 32 are perfect squares, so

Section - 8

Ques 55:

Section - 8

Ans:

Pull out the greatest common perfect square factor 212, leaving perfect-square factors in both parts.

Section - 8

212 and 32 are perfect squares (212 = 26 x 26), so

Section - 8

Ques 56:

Section - 8

Ans:

Pull out the greatest common perfect square factor 502.

Section - 8

Both 502 and 72 are perfect squares, so

Section - 8

The document Formulas & Solved Examples: Exponents & Roots is a part of the CTET & State TET Course Mathematics & Pedagogy Paper 2 for CTET & TET Exams.
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FAQs on Formulas & Solved Examples: Exponents & Roots

1. What is the formula for calculating the value of an exponent?
Ans. The formula for calculating the value of an exponent is given by: a^n = a × a × a × ... (n times), where a is the base number and n is the exponent.
2. How do you simplify expressions with exponents?
Ans. To simplify expressions with exponents, you can use the following rules: - When multiplying terms with the same base, add the exponents: a^m × a^n = a^(m+n) - When dividing terms with the same base, subtract the exponents: a^m ÷ a^n = a^(m-n) - When raising a power to another power, multiply the exponents: (a^m)^n = a^(m × n)
3. What is the formula for calculating the square root of a number?
Ans. The formula for calculating the square root of a number, x, is given by: √x = y, where y is the number that, when multiplied by itself, equals x.
4. How do you simplify expressions with roots?
Ans. To simplify expressions with roots, you can use the following rules: - When multiplying terms with the same root, multiply the numbers inside the root: √(x) × √(y) = √(x × y) - When dividing terms with the same root, divide the numbers inside the root: √(x) ÷ √(y) = √(x ÷ y) - When raising a root to another power, multiply the exponents: (√(x))^n = √(x^n)
5. How do you solve equations involving exponents and roots?
Ans. To solve equations involving exponents and roots, you can follow these steps: 1. Isolate the term with the exponent or root. 2. Apply the appropriate exponent or root operation to both sides of the equation. 3. Simplify the equation using the rules of exponents and roots. 4. Solve for the variable by isolating it on one side of the equation. 5. Check your solution by substituting it back into the original equation.
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