Operations with Scientific Notation

Buster was given this expression and asked to determine whether the answer is greater than or less than 1.

 $\frac{9.763\cdot10^8}{1.56\cdot10^{13}}$  
How can Buster tell without fully performing the calculations? 

• Buster knows the answer is more than 1, because he can add the exponents to see that the solution will have an exponent of 21. 

• Buster knows the answer is less than 1, because he can subtract the exponents to see that the solution will have an exponent of -5. 

• Buster knows the answer is more than 1, because 9.763 / 1.56 is more than 1. 

• Buster cannot answer the question without a calculator. 

Answer explanation

When multiplying numbers in scientific notation, add the exponents to find the power of 10. When dividing numbers in scientific notation, subtract the exponents to find the power of 10. 

Explain the steps you would take to find the product of 6.7 * 10⁶ and 3.2 * 10⁵ written in scientific notation.

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Answer explanation

The coefficient (number being multiplied by the power of 10) must be between 1 and 10. If the number is greater than 10, move the decimal to the left and increase the exponent. If the number is less than 1, move the decimal to the right and decrease the exponent.

Aliyah wants to estimate the value of the given quotient. Which statement about the value is true?  $\frac{5.68\cdot10^{-3}}{8.54\cdot10^{-9}}$  

• The value will be greater than 1 because -3 / -9 is a positive number. A number in scientific notation with a positive exponent has a value greater than 1.

• The value will be greater than 1 because -3 - (-9) = 6. A number in scientific notation with a positive exponent is greater than 1.

• The value will be less than 1 because 5.68 / 8.54 is less than 1. The negative exponents will make it even smaller.

• The number will be less than 1 because 5.68 * 10^-3 is a very small number and division always makes the size of a number smaller.

What is the product in scientific notation? $\left(3.1\cdot10^5\right)\left(2.2\cdot10^7\right)$  

•  $5.5\cdot10^{12}$  
  

•  $6.82\cdot10^{12}$  
  

•  $5.5\cdot10^{35}$  
  

•  $6.82\cdot10^{35}$  
  

Answer explanation

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents.

What is the quotient in scientific notation?

 $\frac{9.3\cdot10^{34}}{3.1\cdot10^{17}}$  

•  $3\cdot10^2$  

•  $3\cdot10^{17}$  

•  $6.2\cdot10^2$  

•  $6.2\cdot10^{17}$  

Answer explanation

When dividing numbers in scientific notation, divide the coefficients and subtract the exponents. 

 $\left(3.25\cdot10^5\right)\left(1.82\cdot10^3\right)$  

When evaluating, what will be the exponent of 10 in the product? 

• 3

• 4

• 8

• 15

 $\frac{6.28\cdot10^{12}}{3.14\cdot10^4}$  

When evaluating, what will be the exponent of 10 in the quotient?

• 3

• 4

• 8

• 16

• He changed the sign of the coefficient. A negative exponent does not affect the sign of a coefficient in scientific notation. The sign of the exponent determines the direction the decimal is moved in.

• He rewrote -28.53 * 10⁴ incorrectly; 28.53 * 10⁴ = 2.853 * 10⁵. The exponent is increased to account for the extra place the decimal is moved.

• He did not correctly evaluate the exponent. It should be evaluated as (10^-7 * 10³) = 10^-21 since exponents are evaluated using the same operation as the coefficients.

• He multiplied the coefficients; he should have added 6.34 and 4.5. The product of powers rule states that coefficients are added.

Which answer shows 35,067,800 written in scientific notation?

• $3.50678\cdot10^{-7}$

• $3.50678\cdot10^7$

• $3.50678\cdot10^8$

• $35067.8\cdot10^2$

Which answer shows 0.00897 written in scientific notation?

• $0.897\cdot10^{-2}$

• $8.97\cdot10^{-3}$

• $8.97\cdot10^{-2}$

• $8.97\cdot10^3$