Standard form questions with answers

Questions

1. Question 1Large numbers

   Express $34\,500\,000$ in standard form.

   Show the worked solution

   1. Step 1: Move the decimal point to just after the first non-zero digitNumber of places the point moves: $7$
   2. Step 2: Write it in standard form$34\,500\,000 = 3.45 \times 10^{7}$
2. Question 2Small numbers

   Express $0.020\,05$ in standard form.

   Show the worked solution

   1. Step 1: Move the decimal point to just after the first non-zero digitNumber of places the point moves: $2$
   2. Step 2: Write it in standard form$0.020\,05 = 2.005 \times 10^{-2}$
3. Question 3Back to an ordinary number

   Write $1.2 \times 10^{3}$ as an ordinary number.

   Show the worked solution

   1. Step 1: Which way does the decimal point move?A positive power of 10 makes the number bigger: the point moves right.
   2. Step 2: Count the places it movesNumber of places the point moves: $3$
   3. Step 3: Write the ordinary number$1.2 \times 10^{3} = 1\,200$
4. Question 4Multiply

   Calculate $(4 \times 10^{-2}) \times (8 \times 10^{6})$, giving your answer in standard form.

   Show the worked solution

   1. Step 1: Deal with the numbers and the powers of 10 separately$(4 \times 8) \times 10^{-2 + 6}$$= 32 \times 10^{4}$
   2. Step 2: Adjust so the number in front is from 1 to less than 10$32 \times 10^{4} = 3.2 \times 10^{5}$
5. Question 5Divide

   Calculate $(4.8 \times 10^{3}) \div (8 \times 10^{2})$, giving your answer in standard form.

   Show the worked solution

   1. Step 1: Deal with the numbers and the powers of 10 separately$(4.8 \div 8) \times 10^{3 - 2}$$= 0.6 \times 10^{1}$
   2. Step 2: Adjust so the number in front is from 1 to less than 10$0.6 \times 10^{1} = 6 \times 10^{0}$
6. Question 6Add and subtract

   Calculate $(8.7 \times 10^{4}) + (1.7 \times 10^{2})$, giving your answer in standard form.

   Show the worked solution

   1. Step 1: Write both with the same power of 10$1.7 \times 10^{2} = 0.017 \times 10^{4}$
   2. Step 2: Add the numbers in front$(8.7 + 0.017) \times 10^{4}$$= 8.717 \times 10^{4}$
7. Question 7A × 10ᵏ for a given k

   Write $8.2 \times 10^{-13}$ in the form $A \times 10^{-12}$. State the value of $A$.

   Show the worked solution

   1. Step 1: Split the power of 10: one part is 10^(-12)$10^{-13} = 10^{-1} \times 10^{-12}$
   2. Step 2: So A is 8.2 times the other part$A = 8.2 \times 10^{-1}$$A = 0.82$
8. Question 8Prefixes: nano, micro, mega, giga

   1 megabyte $(\text{MB}) = 10^{6}$ bytes. Express $2.5$ $\text{MB}$ in bytes, giving your answer in standard form.

   Show the worked solution

   1. Step 1: Replace the prefix mega by its power of 10$2.5\ \text{MB} = 2.5 \times 10^{6}$ bytes
   2. Step 2: Multiplying powers of 10: add the indices (the answer is in bytes)$2.5 \times 10^{6} = 2.5 \times 10^{0 + 6}$$= 2.5 \times 10^{6}$ bytes
9. Question 9Powers of 10 with letters

   It is given that $(4 \times 10^{m}) \times (3 \times 10^{n}) = 1.2 \times 10^{b}$. Express $b$ in terms of $m$ and $n$.

   Show the worked solution

   1. Step 1: Multiply the numbers; add the powers of 10$(4 \times 3) \times 10^{m + n}$$= 12 \times 10^{m + n}$
   2. Step 2: Make the number in front from 1 to less than 10$12 = 1.2 \times 10^{1}$$1.2 \times 10^{1} \times 10^{m + n} = 1.2 \times 10^{m + n + 1}$
   3. Step 3: So b is the whole power of 10$b = m + n + 1$