1.  Write 0.000008 in standard form.

Quick way to do this - count the 0's between the decimal point and the first non-zero digit and add 1 - this give us
5 + 1 = 6

Now, write the number as 8.0 x 10^-6

(the -6 in the power is the 6 you've just worked out with a minus sign in front of it)

Count all the digits after the first number to get the power.

= 4.53 x 10⁸

3.  Change 5.2 x 10⁸ to an ordinary
decimal number.

4. Change 1.65 x 10^-5 to an ordinary decimal number.

5. What is
(2.34 x 10³) x (5.0 x 10⁶) ?

= (2.34 x 5) x (10³ x 10⁶)

Using the rules for indices:
= 11.7 x (10³⁺⁶)

= 11.7 x 10⁹

6. The diameter of the Earth is;
1.27 x 10⁴ km

The diameter of Mars is:
6.79 x 10³ km

What's the difference between the two diameters?

Do this the easiest way as they're quite small numbers, convert back to ordinary decimal numbers to do the subtraction.

12700 - 6790 = 5910

= 5.91 x 10^{3 km}

(using the ordinary decimal numbers it's easy to see that the Earth's diameter is almost twice that of Mars - not so easy to see with standard form)

7.  The mass of an electron is:
0.000 000 000 000 000 000 000 000 91 grams.

How could this mass be written in standard form?

What is the mass of 8,000,000 electrons?

9.1 x 10^-25

8,000,000 = 8 x 10⁶

= (9.1 x 10^-25) x (8 x 10⁶)

= (9.1 x 8) x (10^-25 x 10⁶)

= 72.8 x (10^-25+6)

= 72.8 x 10^-19 grams

= 7.28 x 10^-18 grams (we're only allowed 1 digit in front of the decimal point in standard form)

8. The distance between the Earth and the star Sirius is:
81,900,000,000,000 km

If light travels 3 x 10⁵ km per second, how many seconds does it take for light to travel from Sirius to the Earth?

(give your answer to 2 significant figures)

Distance = 8.19 x 10¹³ km

Speed = Distance / Time, so a bit of algebraic rearrangement gives us:

Time = Distance / Speed

= (8.19 x 10¹³) / (3 x 10⁵)

= (8.19 / 3) x ( 10¹³ - 10⁵)

= 2.73 x 10^13-5

= 2.73 x 10⁸ seconds

= 2.7 x 10⁸ seconds (to 2 significant figures)