Powers of base 10

A power of base ten is a number whose base is 10 raised to an integer power n. It results in the digit 1 followed by n zeros when the exponent is positive, or preceded by n zeros when the exponent is negative.

10 to the power of 0 equals 1 left parenthesis no zero space right parenthesis 10 to the power of 1 equals 10 space left parenthesis a zero space right parenthesis 10 squared equals 100 space left parenthesis two zeros space right parenthesis 10 to the power of right n equals 1 space followed by square space n space zeros

If the exponent n is negative:

10 to the power of minus 1 end of exponential equals space 0 comma 1 space left parenthesis one space preceded space space one space zero right parenthesis 10 to the power of minus 2 end of exponential equals 0 comma 01 space left parenthesis one space preceded space space two space zeros right parenthesis 10 to the power of minus 3 end of exponential equals 0 comma 001 space left parenthesis one space preceded by space of space three space zeros right parenthesis 10 to the power of minus right n end of exponential equals space 1 space preceded right space space n space zeros

In case the exponent is negative, we place a comma after the first zero.

Powers of base ten simplify writing and calculating with large numbers, with many orders or decimal places.

For example, the number 1 000 000 000 (one billion), can be written as 10 to the power of 9 (1 followed by nine zeros). Similarly, a number like 0.000 000 000 001 can be written as 10 to the power of minus to the power of 12 (1 preceded by twelve zeros).

It is worth remembering that this is due to the negative exponent inverting the fraction.

10 to the power of negative squared equals 1 over 10 squared equals 1 over 100 equals 0 point 01

Multiplying and dividing powers of base 10

Multiplication and division of base-ten powers follow the same rules as for power.

When multiplying powers of ten, we repeat the base and add the exponents.

10 to the power of 8. space 10 cubed space equals space 10 to the power of 8 space plus space 3 end of exponential equals 10 to the power of 11

When dividing powers of base 10, we repeat the base and subtract the exponents.

10 to the power of 6 over 10 to the power of 4 equals 10 to the power of 6 minus 4 end of the exponential equals 10 squared

Adding and subtracting powers of base 10

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Addition and subtraction of powers of base ten can only occur if their exponents are equal. Thus, it is enough to treat the powers as integer values.

10 squared plus 10 squared equals 2.10 squared

A power of ten squared plus a power of ten squared is equal to two powers of ten squared.

Example
If the exponents are not equal, equalize them and only then add or subtract.

Changing the exponent in powers of base 10

To change the exponent without changing the value of the power, we multiply the power by 1 and move its decimal point as the exponent changes.

For increase the exponent, we move the decimal point in the digit 1 to the left, we add as many orders as units to the exponent.

Example
Increase the power exponent by 3 10 cubed without changing its value.

10 cubed space equals 1 space. space 10 cubed space equals space 0 comma 001 space. space 10 to the power of 6

For decrease the exponent, we move the decimal point in the digit 1 to the right, as many orders as units we remove from the exponent.

Example
Decrease the power exponent by 2 units 10 to the power of 5 , without changing its value.

10 to the power of 5 equals 1 space. space 10 to the power of 5 space equals space 100 space. space 10 cubed (if we subtract two units in the exponent, we multiply by 100)