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QuestionCalculate 8.12×106×4.7×1038.12 \times 10^{6} \times 4.7 \times 10^{3} using scientific notation. Give your answer with correct significant figures.

Studdy Solution

STEP 1

Assumptions1. We are multiplying two numbers in scientific notation. . The numbers are 8.12×1068.12 \times10^{6} and 4.7×1034.7 \times10^{3}.
3. We will use the product rule for exponents.
4. We will express our answer in scientific notation with the proper number of significant figures.

STEP 2

The product rule for exponents states that when multiplying two powers with the same base, you can add the exponents. So we can separate the multiplication into two parts the multiplication of the significant figures and the multiplication of the powers of ten.
8.12×106×4.7×10=(8.12×4.7)×(106×10)8.12 \times10^{6} \times4.7 \times10^{} = (8.12 \times4.7) \times (10^{6} \times10^{})

STEP 3

Now, calculate the multiplication of the significant figures.
8.12×.7=38.1648.12 \times.7 =38.164

STEP 4

Next, calculate the multiplication of the powers of ten using the product rule for exponents.
106×103=106+3=10910^{6} \times10^{3} =10^{6+3} =10^{9}

STEP 5

Combine the results from steps3 and4 to get the result of the multiplication.
38.164×10938.164 \times10^{9}

STEP 6

However, in scientific notation, the significant figure part should be a number between1 and10. So we need to adjust the result from step5. We can do this by moving the decimal point in38.164 two places to the left, which is equivalent to dividing by100 or multiplying by 10−210^{-2}.
38.164×109=3.8164×101138.164 \times10^{9} =3.8164 \times10^{11}

STEP 7

Finally, we need to express our answer with the proper number of significant figures. Both of our original numbers had three significant figures, so our final answer should also have three significant figures. We round3.8164 to3.82 (since the digit after1 is6 which is greater than5, so we round up).
3.8164×1011=3.82×10113.8164 \times10^{11} =3.82 \times10^{11}So, .12×106.12 \times10^{6} times 4.7×1034.7 \times10^{3} equals 3.82×10113.82 \times10^{11} in scientific notation with the proper number of significant figures.

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