### INDICES & SURDS - The Mistake Fragment part 1

BY MR JOHARI

Context
Below shown the solution posted by the secondary 3 students during the 4th February 2013 Maths Quiz C.
There are errors in the solution posted.

Focus
Error Analysis and Peer Evaluation
Individual Effort

You are required to post the following as a comment:

(i) Nature of error - conceptual or carelessness (be very specific in your description)
example - careless error due to arithmetic (addition)
example - conceptual error due to error in (a + b)^2 = a^2 + b^2.

(2) Correction to the error
Ensure that your solution is concise and correct.

1. Q1) Conceptual error, changed division sign to multiplication sign when changing cube root to fractional exponent)

Solution:
m^x = m^3 / m^1/3 * m^5
m^x = m^7 2/3
x = 7 2/3

Q2a) Conceptual error (32^x ≠ 2^5+x)

Solution:
32x = (2^x)^5 = 3^5 = 243

Q2c) Careless error (3^2 ≠ 6)

Solution:
3/4x = 3/(2^x)^2 = 3/(3)^2 = 1/3

Q3a) Careless error (64 ≠ 2^5)

Solution:
32^x = 64
2^5x = 2^6
x = 1 1/5

Q3c) Careless error (1/16 ≠ 2^-5)

Solution:
32x = 1/16
32x= 16^-1
2^5x = 2^-4
x = -4/5

2. 1) Conceptual error. Changed division sign into multiplication sign after changing cube root into a fractional exponent.

Solution:
m^x = m^3 / m^1/3 * m^5
m^x=m^3-1/3+5
m^x=m^7 2/3
x= 7 2/3

2a) Conceptual Error. 32^x ≠ 2^5+x, 32^x=(2^x)^5

Solution:
32^x=(2^x)^5=(3)^5=243

2c) Careless Error. 3^2 ≠ 6, 3^2=9

Solution:
3/4^x = 3/(2^x)^2 = 3/3^2 = 1/3

3a) Careless Error. 64=2^6, 64≠ 2^5

Solution:
32^x=64
(2^5)^x=2^6
2^5x=2^6
5x=6
x=1 1/5

3c) Careless Error. 1/16 ≠ 2^-5, 1/16=2^-4

32^x=1/16
2^5x=2^-4
5x=-4
x=-4/5

3. Q1. Conceptual error.m^3 X m^(1/3) ≠ m
m^x = m^3 / m^(1/3) X m^5
m^x = m^8/m^(1/3)=m^7 2/3
x=7 2/3

Q2a.Conceptual error 32^x ≠ 2^(5+x)
32x = (2^5)^x = (2^x)^5 = 3^5 = 243

Q2c.Carelessness 3/3^2 ≠ 1/2
3/4^x = 3/(2^x)^2 = 3/(3)^2 = 3/9 = 1/3

3a. Carelessness 2^5 ≠ 64
32^x = 64
2^5x = 2^6
5x = 6
x = 1 1/5

3b.Carelessness 1/16 ≠ 2^-5
32x = 1/16
32x = 2^-4
2^5x = 2^-4
5x = -4
x = -4/5

4. Question 1
Conceptual error due to m^x = m^3 x m^/3 x m^5
m^x = m^3 / m^1/3 x m^5
= m^2 2/3 x m^5
= m^7 2/3
x = 7 2/3

Question 2a
Conceptual error: 32^x ≠ 2^5+x
32^x = 2^5x = (2^x)^5 = 243

Question 2C
Careless error : 3/3^2 ≠ 1/2
3/9 = 1/3

Question 3A
Careless error: 64 ≠ 2^5
32^x = 64
2^5x = 2^6
5x = 6
x = 6/5

Question 3C
Careless error: 1/16 ≠ 2^-5
32^x = 1/16
32^x = 2^-4
5x = -4
x = -4/5

5. 1) Conceptual error. Changed division sign into multiplication sign after changing cube root into a fractional exponent.

Solution:
m^x=m^3/m^1/3*m^5
m^x=m^3-1/3+5
m^x=m^7 2/3
x=7 2/3

2a) Conceptual Error. 32^x ≠ 2^5+x, 32^x=(2^x)^5

Solution:
32^x=(2^x)^5=(3)^5=243

2c) Careless Mistake. 3^2 ≠ 6, 3^2=9

Solution:
3/4^x = 3/(2^x)^2 = 3/3^2 = 1/3

3a) Careless Mistake. 64=2^6, 64≠ 2^5

Solution:
32^x=64
(2^5)^x=2^6
2^5x=2^6
5x=6
x=1 1/5

3c) Careless Error. 1/16 ≠ 2^-5, 1/16=2^-4

32^x=1/16
2^5x=2^-4
5x=-4
x=-4/5

6. 1) Conceptual error

m^x = m^3 / m^1/3 * m^5
=m^3-1/3+5
=m^7 2/3
x= 7 2/3

2a) Conceptual Error
32^x=(2^x)^5
=(3)^5
=243

3a) Careless Error

32^x=64
(2^5)^x=2^6
2^5x=2^6
5x=6
x=6/5

3c) Careless Error

32^x=1/16
2^5x=2^-4
5x=-4
x=-4/5

7. 1. Careless error; changed the division sign to multiplication sign in process of changing the term to indices.

Solution:
m^x = m^3 ÷ m^1/3 x m^5
m^x = m^(7 2/3)
x = 7 2/3

2a. Conceptual error; when expanding 32^x, used multiplication law instead of power law.

Solution:
32^x = (2^5)^x = (2^x)^5 = 3^5 = 243

2c. Careless error; multiplied 3 by 2 in 3^2 instead of 3 to the power of 2.

Solution:
3/4^x = 3/(2^x)^2 = 3/3^2 = 3/9 = 1/3

3a. Careless error; 64 = 2^6 not 2^5.

Solution:
32^x = 64
2^5x = 2^6
5x = 6
x = 6/5
x = 1 1/5

3c. Careless error; 1/16 = 2^-4 not 2^-5.

Solution:
32^x = 1/16
2^5x = 2^-4
5x = -4
x = -4/5

8. 1) Conceptual error: mᵡ = m³ x m^1/3 x m⁵ (division sign changed to multiplication sign after changing cube root to fraction)
solution:
mᵡ = m³ ÷ ∛m x m⁵
mᵡ = m³ ÷ m^1/3 x m⁵
mᵡ = m^5 8/3
∴ x = 7 2/3

2)
a) Conceptual error: 32ᵡ ≠ 2⁵⁺ᵡ
solution:
32ᵡ = 2⁵ᵡ
= (2ᵡ)⁵
= 243

c) Careless error: 3/3² ≠ 1/2
solution:
3/4ᵡ = 3/2²ᵡ
= 3/9
= 1/3

3)
a) Careless error: 64 ≠ 2⁵
solution:
32ᵡ = 64
2⁵ᵡ = 2⁶
∴ 5x = 6
x = 6/5

c) Careless error: 1/16 ≠ 2⁻⁵
32ᵡ = 1/16
32ᵡ = 2⁻⁴
2⁵ᵡ = 2⁻⁴
∴ 5x = -4
x = -4/5

9. Q1: Careless Error (changed division sign to multiplication sign while converting fractional indices)
Conceptual Error ( (m^3) x (m^1/3) ≠ m , should be (m^3) x (m^1/3) = M^3 1/3)
Solution:
m^x = m^3 ÷ m^1/3 x m^5
m^x = (m^2 2/3) x (m^5)
m^x = m^7 2/3
x = 7 2/3

Q2a: Conceptual Error (32^x ≠ 2^x+5 , should be 32^x = 2^5x)
Solution:
32^x = 2^5x
= (2^x)^5
= 3^5
= 243

Q2c: Careless Error (3/3^2 ≠ 1/2, should be 3/3^2 = 1/3)
Solution:
3/4^x = 3/(2^x)^2
= 3/3^2
= 3/9
= 1/3

Q3a: Careless Error (64 ≠ 2^5 , should be 64 = 2^6)
Solution:
32^x = 64
2^5x = 2^6
5x = 6
x = 6/5
x = 1 1/5

Q3c: Careless Error (1/16 ≠ 2^-5 , should be 1/16 = 2^-4)
Solution:
32^x = 1/16
2^5x = 2^-4
5x = -4
x = -4/5

10. 1. Conceptual error
m^x = m^3 / m^(1/3) x m^5
m^x =m^(3-1/3+5)
m^x =m^7 2/3
x= 7 2/3

2a. Conceptual Error
32^x=(2^x)^5
32^x=(3)^5
x=243

3a. Careless
32^x=64
(2^x)^5=2^6
2^5x=2^6
5x=6
x=6/5

3c. Careless
32^x=16^-1
2^5x=2^-4
5x=-4
x=-4/5

11. Questions 2a to 3c I have forgotten how to do therefore I didn't know the answer.. But I have relearnt the steps and redid the quiz on my own and got full marks :OO