Question

the powers of 2 are given in table 1. use this information to answer items 6 to 8.
table 1
exponential form | expanded form | power of 2
$2^{2}$ | $2×2$ | 4
$2^{3}$ | $2×2×2$ | 8
$2^{4}$ | $2×2×2×2$ | 16
$2^{5}$ | $2×2×2×2×2$ | 32

1. show that 1024 is a power of 2. refer to table 1
2. write the exponential form of 1024.
3. find a number that is a power of 2 that meets both of these conditions:

• the number is a multiple of 16.
• the number is also more than 50 but less than 200.

1. is there a number between 0.998 and 0.999? if yes, give one example. if no, explain why you think so.

1. show how you will subtract 0.998 from 0.999.

1. is there a fraction that is greater than $\frac{3}{4}$ but less than 1? if yes, give one example. if no, explain why you think so.

Explanation:

Step1: Relate 1024 to powers of 2

We know from the table that $2^5=32$. We can build up to 1024 by repeatedly multiplying by 2:
$2^6 = 2^5 \times 2 = 32 \times 2 = 64$
$2^7 = 64 \times 2 = 128$
$2^8 = 128 \times 2 = 256$
$2^9 = 256 \times 2 = 512$
$2^{10} = 512 \times 2 = 1024$

Step2: State exponential form of 1024

From Step1, we directly identify the exponential form.

Step3: Find valid power of 2

First, note $16=2^4$. We need a power of 2 greater than 50, less than 200, and a multiple of 16:
$2^6=64$: $64\div16=4$, so it is a multiple of 16, and $50<64<200$.
$2^7=128$: $128\div16=8$, so it is a multiple of 16, and $50<128<200$.
Either is valid, we can choose 64.

Step4: Check number between 0.998 & 0.999

Numbers with more decimal places exist between these two values. An example is 0.9985, which is between the two.

Step5: Subtract 0.998 from 0.999

Align decimal places and perform subtraction:
$$0.999 - 0.998 = 0.001$$

Step6: Find fraction between $\frac{3}{4}$ and 1

$\frac{3}{4}=0.75$. A fraction like $\frac{4}{5}=0.8$, which is greater than 0.75 and less than 1.

Answer:

1. $2^{10}=512\times2=1024$, so 1024 is $2^{10}$, a power of 2.
2. $2^{10}$
3. 64 (or 128)
4. YES, 0.9985
5. $0.999 - 0.998 = 0.001$
6. YES, $\frac{4}{5}$