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Standard form calculations

Standard form calculations

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Summary

Standard form calculations

​​In a nutshell

Standard form is used as an easy way to write large or small numbers. There are rules for adding, subtracting, multiplying and dividing numbers that are written in standard form. 



Writing a number in standard index form

A number is in standard index form if it is written in the form:

a×10ba\times10^b​​


Where the number aa is either an integer or a decimal that is always between 11​ and 1010​, and bb is a whole number that may be negative or positive.


Write a large or small number in standard form

PROCEDURE

​​1.

Move the decimal point until the number is between 11 and 1010​. Count how many spaces the decimal point was moved.

2.

This number between 11​ and 1010​ is the value of aa​.

3.

The number of spaces the decimal point was moved is the value of bb​. If the decimal point moved to the left, then bb​ is positive. If the decimal point was moved to the right, then bb​ is negative.


Example 1

What is 7600076000 in standard index form?


First, write 7600076000 as 76000.0076000.00 and move the decimal point until the number is between 11 and 1010​:

7.6 0 0 0 0 07\overset{\curvearrowleft \curvearrowleft \curvearrowleft \curvearrowleft}{.6\, 0\,0\,0\,}0\thinspace 0​, so a=7.6a=7.6


The decimal point moved 44 spaces to the left, so b=4b=4.

76000=7.6×104\underline{76000=7.6\times10^4}


Example 2

What is 0.00008150.0000815 in standard index form?


First, move the decimal point until the number is between 11 and 1010:

0 0 0 0 0 8.150\thinspace \overset{\curvearrowright \curvearrowright \curvearrowright \curvearrowright \curvearrowright}{0\,0\,0\,0\,8.}15​, so a=8.15a=8.15​​


The decimal point moved 55 spaces to the right, so b=5b=-5.

0.0000815=8.15×105\underline{0.0000815=8.15\times10^{-5}}


Multiplying and dividing in standard index form

To multiply and divide two numbers in standard form, use the rules of indices. Multiply/divide the numbers as normal, and use rules of indices to multiply or divide the powers of 1010. To multiply the powers of 1010, add the powers and to divide powers of 1010, subtract the powers.​


Example 3

What is (1.5×108)×(9×103)(1.5\times10^8)\times(9\times10^{-3})?


Multiply 1.51.5 and 99:

1.5×9=13.51.5 \times 9 = 13.5​​

Use rules of indices to multiply the powers of 1010:

108×103=108+(3)=10510^8 \times 10^{-3} =10^{8+(-3)}= 10^5​​


Combine the answers to give:

(1.5×108)×(9×103)=13.5×105(1.5\times10^8)\times(9\times10^{-3}) = 13.5 \times 10^5​​


Adjust the number so that it is written in standard form. aa should be between 11 and 1010. So change 13.513.5 to 1.351.35 and increase the power of 1010 by 11.


(1.5×108)×(9×103)=1.35×106\underline{(1.5\times10^8)\times(9\times10^{-3}) = 1.35 \times 10^6}


Example 4

What is (8×105)÷(1.25×102)(8\times10^5)\div(1.25\times10^{2})?


Divide 88 by 1.251.25:

8÷1.25=6.48 \div 1.25 = 6.4​​


Use rules of indices to divide the powers of 1010:

105÷102=10310^5 \div 10^{2} = 10^3​​


Combine the answers to give:

(8×105)÷(1.25×102)=6.4×103\underline{(8\times10^5)\div(1.25\times10^{2}) = 6.4 \times 10^3}​​



Adding and subtracting in standard index form

To add and subtract numbers in standard index form, make sure that the powers of 1010 are the same first. Then, add/subtract the numbers and adjust the answer so that the number is between 11​ and 1010​.


Example 5

What is (8×104)+(5×102)(8\times10^4)+(5\times10^2)?


First, make the powers of 1010 the same. The easiest way to do this is to write:

8×104=8×102×102=800×1028\times10^4=8\times10^2\times10^2=800\times10^2​​


Now, add the numbers together:

(800×102)+(5×102)=(800+5)×102=805×102(800\times10^2)+(5\times10^2)=(800+5)\times10^2=805\times10^2​​


Rewrite the number so that the answer is in standard form:

805×102=8.05×102×102=8.05×104805\times10^2=8.05\times10^2\times10^2=8.05\times10^4​​


(8×104)+(5×102)=8.05×104\underline{(8\times10^4)+(5\times10^2)=8.05\times10^4}​​



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