Convert Scientific Notation To Decimal

Learning Outcomes

• Convert from decimal notation to scientific note
• Catechumen from scientific annotation to decimal notation
• Multiply and divide using scientific notation

Catechumen from Decimal Annotation to Scientific Note

Remember working with place value for whole numbers and decimals? Our number organization is based on powers of [latex]10[/latex]. We employ tens, hundreds, thousands, and then on. Our decimal numbers are besides based on powers of tens—tenths, hundredths, thousandths, and and so on.

Consider the numbers [latex]4000[/latex] and [latex]0.004[/latex]. We know that [latex]4000[/latex] means [latex]4\times 1000[/latex] and [latex]0.004[/latex] means [latex]4\times {\Large\frac{1}{1000}}[/latex]. If we write the [latex]1000[/latex] every bit a power of ten in exponential grade, we can rewrite these numbers in this way:

[latex]\brainstorm{array}{cccc}4000\hfill & & & 0.004\hfill \\ 4\times k\hfill & & & 4\times {\Large\frac{1}{1000}}\hfill \\ iv\times {x}^{3}\hfill & & & 4\times {\Large\frac{1}{{10}^{3}}}\hfill \\ & & & \hfill four\times {10}^{-3}\hfill \end{array}[/latex]

When a number is written every bit a product of two numbers, where the starting time factor is a number greater than or equal to ane just less than [latex]10[/latex], and the second factor is a power of [latex]x[/latex] written in exponential form, it is said to be in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is of the form

[latex]a\times {10}^{northward}[/latex]

where [latex]a\ge 1[/latex] and [latex]a<10[/latex] and [latex]n[/latex] is an integer.

It is customary in scientific notation to use [latex]\times [/latex] equally the multiplication sign, even though we avert using this sign elsewhere in algebra.

Scientific annotation is a useful manner of writing very large or very minor numbers. It is used often in the sciences to brand calculations easier.

If nosotros look at what happened to the decimal betoken, we can meet a method to easily convert from decimal notation to scientific note.

In both cases, the decimal was moved [latex]3[/latex] places to get the start factor, [latex]four[/latex], by itself.

• The ability of [latex]ten[/latex] is positive when the number is larger than [latex]one\text{ : }4000=four\times {10}^{3}[/latex].
• The power of [latex]x[/latex] is negative when the number is betwixt [latex]0[/latex] and [latex]i\text{ : }0.004=4\times {10}^{-three}[/latex].

example

Write [latex]37,000[/latex] in scientific notation.

Solution

Step 1: Move the decimal signal so that the first factor is greater than or equal to [latex]1[/latex] but less than [latex]10[/latex].
Step 2: Count the number of decimal places, [latex]north[/latex] , that the decimal point was moved.	[latex]3.70000[/latex] [latex]4[/latex] places
Stride 3: Write the number as a production with a power of [latex]x[/latex].	[latex]3.seven\times {ten}^{4}[/latex]
If the original number is: greater than i, the power of 10 will be [latex]{10}^{north}[/latex] . between 0 and i, the power of ten will be [latex]{10}^{\mathrm{-due north}}[/latex]
Step 4: Check.
[latex]{10}^{4}[/latex] is [latex]10,000[/latex] and [latex]10,000[/latex] times [latex]three.seven[/latex] will be [latex]37,000[/latex].
	[latex]37,000=three.vii\times {10}^{four}[/latex]

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Catechumen from decimal notation to scientific annotation

1. Movement the decimal point so that the first cistron is greater than or equal to [latex]1[/latex] but less than [latex]10[/latex].
2. Count the number of decimal places, [latex]north[/latex], that the decimal signal was moved.
3. Write the number equally a product with a power of [latex]ten[/latex].
   • If the original number is:
     • greater than [latex]1[/latex], the power of [latex]10[/latex] will exist [latex]{10}^{northward}[/latex].
     • betwixt [latex]0[/latex] and [latex]i[/latex], the power of [latex]ten[/latex] will be [latex]{x}^{-n}[/latex].
4. Check.

example

Write in scientific notation: [latex]0.0052[/latex]

endeavour it

Convert Scientific Notation to Decimal Form

How can we catechumen from scientific notation to decimal grade? Let's look at 2 numbers written in scientific notation and come across.

[latex]\begin{array}{cccc}9.12\times {x}^{4}\hfill & & & ix.12\times {x}^{-iv}\hfill \\ nine.12\times 10,000\hfill & & & 9.12\times 0.0001\hfill \\ 91,200\hfill & & & 0.000912\hfill \stop{array}[/latex]

If we expect at the location of the decimal point, we can see an easy method to catechumen a number from scientific notation to decimal form.

In both cases the decimal signal moved 4 places. When the exponent was positive, the decimal moved to the correct. When the exponent was negative, the decimal betoken moved to the left.

example

Catechumen to decimal form: [latex]6.2\times {ten}^{3}[/latex]

effort it

Convert scientific annotation to decimal form

1. Determine the exponent, [latex]north[/latex], on the factor [latex]x[/latex].
2. Motion the decimal [latex]northward[/latex] places, adding zeros if needed.
   • If the exponent is positive, motion the decimal point [latex]n[/latex] places to the right.
   • If the exponent is negative, move the decimal signal [latex]|n|[/latex] places to the left.
3. Check.

example

Convert to decimal form: [latex]8.9\times {10}^{-2}[/latex]

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Multiply and Divide Using Scientific Annotation

We use the Properties of Exponents to multiply and divide numbers in scientific note.

instance

Multiply. Write answers in decimal grade: [latex]\left(4\times {10}^{5}\right)\left(ii\times {ten}^{-7}\right)[/latex].

try it

example

Divide. Write answers in decimal form: [latex]{\Large\frac{9\times {10}^{3}}{3\times {10}^{-2}}}[/latex].

try it

The following video is a mini-lesson on how to convert decimals to scientific notation, and back to a decimal. Additionally, y'all volition see more examples of how to multiply and split numbers given in scientific annotation.

Convert Scientific Notation To Decimal,

Source: https://courses.lumenlearning.com/monroecc-prealgebra/chapter/scientific-notation/

Posted by: danielssoing1993.blogspot.com

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