Simplifications

Problem 1: Simplify (33076161)^⅓ + (279841)^¼ + (243)^⅕ = ?

(33076161)^1/3 + (279841)^1/4 + (243)^1/5 = ?

Given options are

  • 344
  • 345
  • 346
  • 347
  • None of these

The solution involves four steps

  • Finding the cube root of the number 33076161
  • Finding the fourth root of 279841
  • Find the fifth root of 243
  • Adding all the three numbers
  1. Finding the cube root of the number 33076161

TP SP FP – We will be finding the numbers in these three places as below.

  • Divide the number into three parts
    • First number (FN) : 33
    • Second number (SN): 076
    • The third number (TN): 161

To find the cube root, we will use TN, then FN and finally SN.

  • TN ends with 1, so the number in the one’s place (FP) is 1. (Refer Table 1 below)

TP SP 1

  • FN is 33, which in between the cube of 3 and 4, so we have to choose the small number which is 3 will be in third place (TP). (Refer table 1 below)

3 SP 1

  • To find the number in second place, we can use the formula – 3*(TP^2)*SP= second digit of (TN-FP)
  • 3*(1^2)*SP= Second digit of (161-1) = Second digit of 160 = 6
  • 3*(1^2)*SP=6
  • SP=6/3=2

3 2 1

  • Finally, the cube root of (33076161)⅓ is 321

2. Finding the fourth root of 279841

  • First, find the square root of the number 279841
  • The square root of 279841 is 529
  • Again find the square root of 529
  • The square root of 529 is 23
  • (23^4)¼=23

3. Find the fifth root of 243

  • Rewrite 243 as 3^5
  • Cancel the common factor of 5
  • And the answer is 3

4. Then, the final step is adding all the three numbers 321+23+3=347

Table 1: Square, Cube, Numbers Ranging 0 – 10

Number x

Square x2

Cube x3

1

1

1

2

4

8

3

9

27

4

16

64

5

25

125

6

36

216

7

49

343

8

64

512

9

81

729

10

100

1000

Problem2:  55555+5555+555+55+5+=?

To find the solution in an easy way first we have to count the no.of digits in a number and we have to choose the highest digit number

Step1: In this number 55555, the no.of digits are 5

5555=4

555=3

55=2

5=1

Step 2: Multiply as shown in the below

5*5=25 here 5 is the number and 2 is carried

5*4=20 add the previous carry to the number 20 then 20+2=22 2 is the number and 2 is carried

5*3=15 add the previous carry to the number 15 then 15+2=17 7 is the number and 1 is carried

5*2=10 add the previous carry to the number 10 then 10+1=11 1 is the number and 1 is carried

5*1=5 add the previous carry to the number 10 then 5+1=6 so 6 is the number

And finally, the answer is 61725.

Number

digit* no.of digits

sum

Carry Forward (CF)

sum+CF

Final Digit

55555

5*5

25

0

25

5

5555

5*4

20

2

22

2

555

5*3

15

2

17

7

55

5*2

10

1

11

1

5

5*1

5

1

6

6

Final Digit = second digit of (sum+CF)

Carry Forward = first digit of previous (sum+CF)

So the Final sum is 61725

 

Problem 3: 0.4444+0.444+0.44+0.4=?

To find the solution for this there is a shortcut which is similar to the above problem

Here we have to choose the smallest digit number

The smallest digit is 0.4 the no.of digits are 1

0.44=2

0.444=3

0.4444=4

Step 2: multiply as shown in the below

4*1=4

4*2=8

4*3=12 here 2 is the number and 1 is carried

4*4=16 add the previous carry to the number 16 then 16+1=17

And finally, we get 17284

After four digits there is a decimal point so the answer is 1.7284

Number

digit* no.of digits

sum

Carry Forward (CF)

sum+CF

Final Digit

0.4

4*1

4

0

4

4

0.44

4*2

8

0

8

8

0.444

4*3

12

1

2

2

0.4444

4*4

16

1

17

17

Final Digit = Second digit of (sum+CF)

Carry Forward = First digit of previous (sum+CF)

So the Final sum is 1.7284

 

Problem 4: 3333.666+33.6666+333.66+3.6=?

This can be divided into two parts

(3333+33+333+3)+(0.666+0.6666+0.66+0.6)

4            2      3      1          3             4             2        1

Part-1 and 2 can solve by using the above method

Choose highest digit number choose the lowest digit number

3*4=12 2 is number 1 is carry 6*1=6

3*3=9 9+1=10 0 is number 1 is carry 6*2=12 2 is number and 1 is carry

3*2=6 6+1=7 6*3=18 18+1=19 9 is the number 1 is carry

3*1=3 6*4=24 24+1=25

Ans is 3702

ans is 2.5926

3702+2.5926=3704.5926

 

Trick: 61+16

When it is addition we have to multiply with 11

11*(1+6) or 11*(6+1)

11*7=77

Trick: 76-67

 

When it is subtraction we have to multiply with 9

9(7-6)

9*1=9

 

Problem: 53+35-56-65+63+36-98-89=?

53+35-(56+65)+63+36-(98+89)

11(8-11+9-17)

11*-11

-121

Problem: 53-35+83-38+68-86=?

9(2+5-2)

9*5=45

Trick: 65*75

1.7+1=8

2.6*8=48

3.difference between 6 and 7 is 1

4.last digit is 25

5.4825+1*50

=4825+50

=4875

Example: 25*95

9+1=10

2*10=20

2025

Difference between 2 and 9 is 7

2025+7*50

2025+350

2375