Solve problems using scientific notation - HiSet: High School Equivalency Test: Math

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Question

Simplify the following expression using scientific notation.

$(1.26\times10^6)+(2.4\times10^4)$

Answer

You can solve this problem in several ways. One way is to convert each number out of scientific notation and write it out fully, then find the sum of the two values and convert the answer back into scientific notation.

$1.26\times10^6+2.4\times10^4 ewline = 1,260,000+24,000 ewline = 1,284,000 ewline = 1.284\times10^6$

Another, potentially faster, way to solve this problem is to convert one answer into the same scientific-notational terms as the other and then sum them.

$1.26\times10^6+2.4\times10^4$

$2.4\times10^{4}=0.024\times10^{6}$

$1.26\times10^6+0.024\times10^6=(1.26+0.024)\times10^6=1.284\times10^6$

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Question

Multiply, and express the product in scientific notation:

$7,200,000 \times 5,000,000$

Answer

Convert 7,200,000 to scientific notation as follows:

Move the (implied) decimal point until it is immediately after the first nonzero digit (the 7). This required moving the point six units to the left:

$7,200,000. \Rightarrow 7.200000$

It follows that 7,200,000 can be rewritten as $7.2 \times 10^{6}$ .

By similar reasoning, 5,000,000 can be rewritten as $5 \times 10^{6}$ .

Thus,

$7,200,000 \times 5,000,000$

$= (7.2\times 10 ^{6} )\times (5 \times 10^{6} )$

Rearrange and regroup the expressions so that the powers of ten are together:

$= (7.2\times 5) \times (10 ^{6} \times 10^{6} )$

Multiply the numbers in front. Also, multiply the powers of ten by adding exponents:

$= 36 \times 10 ^{6+6}$

$= 36 \times 10 ^{12}$

In order for the number to be in scientific notation, the number in front must be between 1 and 10. An adjustment must be made by moving the implied decimal point in 36 one unit left. It follows that

$36 = 3.6 \times 10^{1}$

Therefore,

$36 \times 10 ^{12} = (3.6 \times 10^{1}) \times 10 ^{12}$

$= 3.6 \times 10^{1+12}$

$= 3.6 \times 10^{13}$ ,

the correct response.

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Question

Multiply:

$5,000,000 \times 120,000,000$

Express the product in scientific notation.

Answer

Scientific notation refers to a number expressed in the form

$\pm a \times 10^{N}$ ,

where $1 \le a < 10$ and is an integer.

Each factor can be rewritten in scientific notation as follows:

$5,000,000 = 5 \times 1,000,000 = 5 \times 10^{6}$

$120,000,000 = 1.2 \times 100,000,000 = 1.2 \times 10^{8}$

Now, substitute:

$5,000,000 \times 120,000,000$

$= 5 \times 10^{6} \times 1.2 \times 10^{8}$

$=( 5\times 1.2) \times (10^{6} \times 10^{8})$

$= 6 \times 10^{6} \times 10^{8}$

Apply the Product of Powers Property:

$= 6 \times 10^{6+8}$

$= 6 \times 10^{14}$

This is in scientific notation and is the correct choice.

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