express each answer in exponential form using the indicated bases:
1. 4 x 4 x 4 x 4, base 2
2. 100 x 100 x 100, bases 2 and 5
3. 4 x 8 x 4 x 8, base 4
I will do #2 for you
100x100x100
= (4x25)^3
= (2^2 x 5^2)^3
= (2^6)(5^6)
let me know what you got for the other two.
100x3
1. To express the answer, 4 x 4 x 4 x 4, in base 2, we need to determine the value of 4 raised to the power of 4 and then convert it to base 2.
4 raised to the power of 4 is written as 4^4.
We can calculate 4^4 as follows:
4^4 = 4 x 4 x 4 x 4 = 256.
Now, let's convert 256 to base 2:
In base 2, the values range from 0 to 1, so we divide the number by 2 and record the remainder:
256 ÷ 2 = 128 with a remainder of 0
128 ÷ 2 = 64 with a remainder of 0
64 ÷ 2 = 32 with a remainder of 0
32 ÷ 2 = 16 with a remainder of 0
16 ÷ 2 = 8 with a remainder of 0
8 ÷ 2 = 4 with a remainder of 0
4 ÷ 2 = 2 with a remainder of 0
2 ÷ 2 = 1 with a remainder of 0
1 ÷ 2 = 0 with a remainder of 1
Now, we write the remainders (from bottom to top) to express the answer in base 2:
256 in base 10 is equivalent to 100000000 in base 2.
Therefore, the answer to 1. in base 2 is 4 x 4 x 4 x 4 = 100000000.
2. To express the answer, 100 x 100 x 100, in bases 2 and 5, we need to determine the values of 100^3 and then convert them to bases 2 and 5.
100 raised to the power of 3 is written as 100^3.
We can calculate 100^3 as follows:
100^3 = 100 x 100 x 100 = 1,000,000.
Now, let's convert 1,000,000 to base 2:
1,000,000 ÷ 2 = 500,000 with a remainder of 0
500,000 ÷ 2 = 250,000 with a remainder of 0
250,000 ÷ 2 = 125,000 with a remainder of 0
125,000 ÷ 2 = 62,500 with a remainder of 0
62,500 ÷ 2 = 31,250 with a remainder of 0
31,250 ÷ 2 = 15,625 with a remainder of 0
15,625 ÷ 2 = 7,812.5 with a remainder of 1
7,812 ÷ 2 = 3,906 with a remainder of 0
3,906 ÷ 2 = 1,953 with a remainder of 0
1,953 ÷ 2 = 976.5 with a remainder of 1
976 ÷ 2 = 488 with a remainder of 0
488 ÷ 2 = 244 with a remainder of 0
244 ÷ 2 = 122 with a remainder of 0
122 ÷ 2 = 61 with a remainder of 0
61 ÷ 2 = 30.5 with a remainder of 1
30 ÷ 2 = 15 with a remainder of 0
15 ÷ 2 = 7.5 with a remainder of 1
7 ÷ 2 = 3.5 with a remainder of 1
3 ÷ 2 = 1.5 with a remainder of 1
1 ÷ 2 = 0.5 with a remainder of 1
Now, we write the remainders (from bottom to top) to express the answer in base 2:
1,000,000 in base 10 is equivalent to 11110100001001000000 in base 2.
Now, let's convert 1,000,000 to base 5:
1,000,000 ÷ 5 = 200,000 with a remainder of 0
200,000 ÷ 5 = 40,000 with a remainder of 0
40,000 ÷ 5 = 8,000 with a remainder of 0
8,000 ÷ 5 = 1,600 with a remainder of 0
1,600 ÷ 5 = 320 with a remainder of 0
320 ÷ 5 = 64 with a remainder of 0
64 ÷ 5 = 12 with a remainder of 4
12 ÷ 5 = 2 with a remainder of 2
2 ÷ 5 = 0 with a remainder of 2
Now, we write the remainders (from bottom to top) to express the answer in base 5:
1,000,000 in base 10 is equivalent to 20000002 in base 5.
Therefore, the answers to 2. in bases 2 and 5 are:
- In base 2: 100 x 100 x 100 = 11110100001001000000.
- In base 5: 100 x 100 x 100 = 20000002.
3. To express the answer, 4 x 8 x 4 x 8, in base 4, we need to determine the value of 4 raised to the power of 4 and then convert it to base 4.
4 raised to the power of 4 is written as 4^4.
We can calculate 4^4 as follows:
4^4 = 4 x 4 x 4 x 4 = 256.
Now, let's convert 256 to base 4:
256 ÷ 4 = 64 with a remainder of 0
64 ÷ 4 = 16 with a remainder of 0
16 ÷ 4 = 4 with a remainder of 0
4 ÷ 4 = 1 with a remainder of 0
1 ÷ 4 = 0 with a remainder of 1
Now, we write the remainders (from bottom to top) to express the answer in base 4:
256 in base 10 is equivalent to 10000 in base 4.
Therefore, the answer to 3. in base 4 is 4 x 8 x 4 x 8 = 10000.
To express each answer in exponential form, we'll use the bases indicated.
1. 4 x 4 x 4 x 4, base 2:
To express this in exponential form, we need to determine the power to which the base is raised to obtain the product.
In this case, the base is 4, and we want to express the product in base 2. To do this, we need to convert the base 4 to base 2.
To convert from base 4 to base 2, we can rewrite 4 as 2^2. Therefore, we have:
(2^2) x (2^2) x (2^2) x (2^2)
Now, when multiplying the same base with different exponents, we can add the exponents. Therefore, our exponential form is:
2^(2 + 2 + 2 + 2) = 2^8
So, the answer in exponential form, using the base indicated, is 2^8.
2. 100 x 100 x 100, bases 2 and 5:
To express this product in exponential form, we'll do it separately for bases 2 and 5.
For base 2:
Since the base is 100, and we want to express the product in base 2, we need to convert 100 to base 2.
To convert from base 100 to base 2, we can rewrite 100 as (10^2)^2, which gives us:
(10^2)^2 x (10^2)^2 x (10^2)^2
Similar to the previous example, when multiplying the same base with different exponents, we add the exponents. Therefore, our exponential form is:
10^(2 + 2 + 2 + 2) = 10^8
Now, we need to convert 10 to base 2. To do this, we can break down 10 into 2 x 5:
(2 x 5)^8
Using the property of exponents, we can distribute the exponent to both 2 and 5:
2^8 x 5^8
Therefore, the answer in exponential form, using base 2, is 2^8 x 5^8.
For base 5:
Continuing from the previous step, we need to express 100 in base 5.
We can write 100 as 5^2 x 4:
(5^2 x 4)^8
Again, using the property of exponents, we can distribute the exponent to both 5^2 and 4:
5^(2 x 8) x 4^8
Therefore, the answer in exponential form, using base 5, is 5^(2 x 8) x 4^8.
3. 4 x 8 x 4 x 8, base 4:
To express this product in exponential form, we need to convert the base from 4 to 4.
To do this, we can rewrite 4 as 4^1:
(4^1) x (8) x (4^1) x (8)
Again, when multiplying the same base with different exponents, we can add the exponents. Therefore, our exponential form is:
4^(1 + 1 + 1 + 1)
Simplifying the exponent gives us:
4^4
So, the answer in exponential form, using base 4, is 4^4.