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Math  /  Numbers & Operations

QuestionExpress the product and the quotients of the following with the appropriate number of sig figs: 1) (3.56×105)(4.21×106)=\left(3.56 \times 10^{5}\right)\left(4.21 \times 10^{6}\right)= 2) (2×107)(8×10−9)=\left(2 \times 10^{7}\right)\left(8 \times 10^{-9}\right)= 3) (4.11×10−5)(7.51×10−4)=\left(4.11 \times 10^{-5}\right)\left(7.51 \times 10^{-4}\right)= 4) 8.45×107/6.74×103=8.45 \times 10^{7} / 6.74 \times 10^{3}= 5) 9.7×108/8.6×10−2=9.7 \times 10^{8} / 8.6 \times 10^{-2}= 6) 4.7×10−2/5.7×10−6=4.7 \times 10^{-2} / 5.7 \times 10^{-6}=

Studdy Solution

STEP 1

1. The significant figures (sig figs) in the given numbers must be preserved throughout the calculations.
2. Multiplication and division rules for significant figures must be applied.
3. The final result should be expressed with the appropriate number of significant figures based on the given data.

STEP 2

1. Calculate the product of (3.56×105)\left(3.56 \times 10^{5}\right) and (4.21×106)\left(4.21 \times 10^{6}\right) with the correct number of significant figures.
2. Calculate the product of (2×107)\left(2 \times 10^{7}\right) and (8×10−9)\left(8 \times 10^{-9}\right) with the correct number of significant figures.
3. Calculate the product of (4.11×10−5)\left(4.11 \times 10^{-5}\right) and (7.51×10−4)\left(7.51 \times 10^{-4}\right) with the correct number of significant figures.
4. Calculate the quotient of 8.45×1078.45 \times 10^{7} divided by 6.74×1036.74 \times 10^{3} with the correct number of significant figures.
5. Calculate the quotient of 9.7×1089.7 \times 10^{8} divided by 8.6×10−28.6 \times 10^{-2} with the correct number of significant figures.
6. Calculate the quotient of 4.7×10−24.7 \times 10^{-2} divided by 5.7×10−65.7 \times 10^{-6} with the correct number of significant figures.

STEP 3

Multiply the coefficients of (3.56×105)\left(3.56 \times 10^{5}\right) and (4.21×106)\left(4.21 \times 10^{6}\right).
3.56×4.21=14.9876 3.56 \times 4.21 = 14.9876

STEP 4

Determine the number of significant figures in the product. Both numbers have 3 significant figures.
Round 14.9876 to 3 significant figures: 15.0 \text{Round } 14.9876 \text{ to 3 significant figures: } 15.0

STEP 5

Multiply the powers of 10:
105×106=1011 10^{5} \times 10^{6} = 10^{11}

STEP 6

Combine the results:
(3.56×105)×(4.21×106)=1.50×1012 (3.56 \times 10^{5}) \times (4.21 \times 10^{6}) = 1.50 \times 10^{12}

STEP 7

Multiply the coefficients of (2×107)\left(2 \times 10^{7}\right) and (8×10−9)\left(8 \times 10^{-9}\right).
2×8=16 2 \times 8 = 16

STEP 8

Determine the number of significant figures in the product. The least number of significant figures is 1.
Round 16 to 1 significant figure: 2×101 \text{Round } 16 \text{ to 1 significant figure: } 2 \times 10^{1}

STEP 9

Multiply the powers of 10:
107×10−9=10−2 10^{7} \times 10^{-9} = 10^{-2}

STEP 10

Combine the results:
(2×107)×(8×10−9)=2×10−1 (2 \times 10^{7}) \times (8 \times 10^{-9}) = 2 \times 10^{-1}

STEP 11

Multiply the coefficients of (4.11×10−5)\left(4.11 \times 10^{-5}\right) and (7.51×10−4)\left(7.51 \times 10^{-4}\right).
4.11×7.51=30.8661 4.11 \times 7.51 = 30.8661

STEP 12

Determine the number of significant figures in the product. Both numbers have 3 significant figures.
Round 30.8661 to 3 significant figures: 30.9 \text{Round } 30.8661 \text{ to 3 significant figures: } 30.9

STEP 13

Multiply the powers of 10:
10−5×10−4=10−9 10^{-5} \times 10^{-4} = 10^{-9}

STEP 14

Combine the results:
(4.11×10−5)×(7.51×10−4)=3.09×10−8 (4.11 \times 10^{-5}) \times (7.51 \times 10^{-4}) = 3.09 \times 10^{-8}

STEP 15

Divide the coefficients of 8.45×1078.45 \times 10^{7} by 6.74×1036.74 \times 10^{3}.
8.456.74=1.2537 \frac{8.45}{6.74} = 1.2537

STEP 16

Determine the number of significant figures in the quotient. The least number of significant figures is 3.
Round 1.2537 to 3 significant figures: 1.25 \text{Round } 1.2537 \text{ to 3 significant figures: } 1.25

STEP 17

Divide the powers of 10:
107/103=104 10^{7} / 10^{3} = 10^{4}

STEP 18

Combine the results:
8.45×1076.74×103=1.25×104 \frac{8.45 \times 10^{7}}{6.74 \times 10^{3}} = 1.25 \times 10^{4}

STEP 19

Divide the coefficients of 9.7×1089.7 \times 10^{8} by 8.6×10−28.6 \times 10^{-2}.
9.78.6=1.1279 \frac{9.7}{8.6} = 1.1279

STEP 20

Determine the number of significant figures in the quotient. The least number of significant figures is 2.
Round 1.1279 to 2 significant figures: 1.1 \text{Round } 1.1279 \text{ to 2 significant figures: } 1.1

STEP 21

Divide the powers of 10:
108/10−2=1010 10^{8} / 10^{-2} = 10^{10}

STEP 22

Combine the results:
9.7×1088.6×10−2=1.1×1010 \frac{9.7 \times 10^{8}}{8.6 \times 10^{-2}} = 1.1 \times 10^{10}

STEP 23

Divide the coefficients of 4.7×10−24.7 \times 10^{-2} by 5.7×10−65.7 \times 10^{-6}.
4.75.7=0.8246 \frac{4.7}{5.7} = 0.8246

STEP 24

Determine the number of significant figures in the quotient. The least number of significant figures is 2.
Round 0.8246 to 2 significant figures: 0.82 \text{Round } 0.8246 \text{ to 2 significant figures: } 0.82

STEP 25

Divide the powers of 10:
10−2/10−6=104 10^{-2} / 10^{-6} = 10^{4}

STEP 26

Combine the results:
4.7×10−25.7×10−6=8.2×102 \frac{4.7 \times 10^{-2}}{5.7 \times 10^{-6}} = 8.2 \times 10^{2}
Solution Summary: 1) (3.56×105)(4.21×106)=1.50×1012(3.56 \times 10^{5})(4.21 \times 10^{6}) = 1.50 \times 10^{12} 2) (2×107)(8×10−9)=2×10−1(2 \times 10^{7})(8 \times 10^{-9}) = 2 \times 10^{-1} 3) (4.11×10−5)(7.51×10−4)=3.09×10−8(4.11 \times 10^{-5})(7.51 \times 10^{-4}) = 3.09 \times 10^{-8} 4) 8.45×1076.74×103=1.25×104\frac{8.45 \times 10^{7}}{6.74 \times 10^{3}} = 1.25 \times 10^{4} 5) 9.7×1088.6×10−2=1.1×1010\frac{9.7 \times 10^{8}}{8.6 \times 10^{-2}} = 1.1 \times 10^{10} 6) 4.7×10−25.7×10−6=8.2×102\frac{4.7 \times 10^{-2}}{5.7 \times 10^{-6}} = 8.2 \times 10^{2}

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