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QuestionCalculate these operations in the given bases: a. 31five⋅3five31_{\text{five}} \cdot 3_{\text{five}} b. 31five+3five31_{\text{five}} + 3_{\text{five}} c. 32six⋅23six32_{\text{six}} \cdot 23_{\text{six}} d. 124five+3five124_{\text{five}} + 3_{\text{five}} e. 10100two+10two10100_{\text{two}} + 10_{\text{two}} f. 10011two⋅100two10011_{\text{two}} \cdot 100_{\text{two}}

Studdy Solution

STEP 1

Assumptions1. All numbers are given in base notation. . The operations are multiplication, addition, and division.
3. The bases are five, six, and two.
4. The results are given in the same base as the operands.

STEP 2

First, let's solve the first operation 31five ⋅five 31_{\text {five }} \cdot_{\text {five }}. To do this, we need to convert the numbers to base ten, perform the multiplication, and then convert the result back to base five.
31five =⋅51+1⋅50=15+1=16ten 31_{\text {five }} = \cdot5^1 +1 \cdot5^0 =15 +1 =16_{\text {ten }}five =⋅50=ten _{\text {five }} = \cdot5^0 =_{\text {ten }}

STEP 3

Perform the multiplication in base ten.
16ten ⋅3ten =48ten 16_{\text {ten }} \cdot3_{\text {ten }} =48_{\text {ten }}

STEP 4

Convert the result back to base five.
48ten =1⋅3+4⋅1+3⋅0=143five 48_{\text {ten }} =1 \cdot^3 +4 \cdot^1 +3 \cdot^0 =143_{\text {five }}

STEP 5

Next, let's solve the second operation 31five +3five 31_{\text {five }}+3_{\text {five }}. To do this, we need to convert the numbers to base ten, perform the addition, and then convert the result back to base five.
31five =3⋅51+1⋅50=15+1=16ten 31_{\text {five }} =3 \cdot5^1 +1 \cdot5^0 =15 +1 =16_{\text {ten }}3five =3⋅50=3ten 3_{\text {five }} =3 \cdot5^0 =3_{\text {ten }}

STEP 6

Perform the addition in base ten.
16ten +3ten =19ten 16_{\text {ten }} +3_{\text {ten }} =19_{\text {ten }}

STEP 7

Convert the result back to base five.
19ten =3⋅51+4⋅50=34five 19_{\text {ten }} =3 \cdot5^1 +4 \cdot5^0 =34_{\text {five }}

STEP 8

Now, let's solve the third operation 32six ⋅23six 32_{\text {six }} \cdot23_{\text {six }}. To do this, we need to convert the numbers to base ten, perform the multiplication, and then convert the result back to base six.
32six =3⋅61+2⋅60=18+2=20ten 32_{\text {six }} =3 \cdot6^1 +2 \cdot6^0 =18 +2 =20_{\text {ten }}23six =2⋅61+3⋅60=12+3=15ten 23_{\text {six }} =2 \cdot6^1 +3 \cdot6^0 =12 +3 =15_{\text {ten }}

STEP 9

Perform the multiplication in base ten.
20ten ⋅15ten =300ten 20_{\text {ten }} \cdot15_{\text {ten }} =300_{\text {ten }}

STEP 10

Convert the result back to base six.
300ten =2⋅63+2⋅62+0⋅6+0⋅60=2200six 300_{\text {ten }} =2 \cdot6^3 +2 \cdot6^2 +0 \cdot6^ +0 \cdot6^0 =2200_{\text {six }}

STEP 11

Finally, let's solve the last operation 10011two ⋅100two 10011_{\text {two }} \cdot100_{\text {two }}. To do this, we need to convert the numbers to base ten, perform the multiplication, and then convert the result back to base two.
10011two =⋅4+0⋅3+0⋅+⋅+⋅0=16++=19ten 10011_{\text {two }} = \cdot^4 +0 \cdot^3 +0 \cdot^ + \cdot^ + \cdot^0 =16 + + =19_{\text {ten }}100two =⋅+0⋅+0⋅0=4ten 100_{\text {two }} = \cdot^ +0 \cdot^ +0 \cdot^0 =4_{\text {ten }}

STEP 12

Perform the multiplication in base ten.
19ten ⋅4ten =76ten 19_{\text {ten }} \cdot4_{\text {ten }} =76_{\text {ten }}

STEP 13

Convert the result back to base two.
76ten =⋅26+0⋅25+0⋅2+⋅23+⋅22+0⋅2+0⋅20=1001100two 76_{\text {ten }} = \cdot2^6 +0 \cdot2^5 +0 \cdot2^ + \cdot2^3 + \cdot2^2 +0 \cdot2^ +0 \cdot2^0 =1001100_{\text {two }}So, the solutions area. 31five ⋅3five =143five 31_{\text {five }} \cdot3_{\text {five }}=143_{\text {five }} b. 31five +3five =34five 31_{\text {five }}+3_{\text {five }}=34_{\text {five }} c. 32six ⋅23six =2200six 32_{\text {six }} \cdot23_{\text {six }}=2200_{\text {six }} f. 10011two ⋅100two =1001100two 10011_{\text {two }} \cdot100_{\text {two }}=1001100_{\text {two }}

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