#### 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

- 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