Answer :
To determine which expression is equivalent to [tex]\(2(5)^4\)[/tex], we need to evaluate each of the given expressions and compare them to the value of [tex]\(2(5)^4\)[/tex].
First, let's calculate [tex]\(2(5)^4\)[/tex]:
- [tex]\(5^4\)[/tex] means multiplying 5 by itself 4 times: [tex]\(5 \times 5 \times 5 \times 5\)[/tex].
- Now, let's calculate [tex]\(5 \times 5 \times 5 \times 5\)[/tex]:
[tex]\[ 5 \times 5 = 25 \][/tex]
[tex]\[ 25 \times 5 = 125 \][/tex]
[tex]\[ 125 \times 5 = 625 \][/tex]
- So, [tex]\(5^4 = 625\)[/tex].
- Now, multiply this result by 2:
[tex]\[ 2 \times 625 = 1250 \][/tex]
Thus, [tex]\(2(5)^4 = 1250\)[/tex].
Next, let's evaluate each of the given expressions and see which one equals 1250:
1. [tex]\(2 \times 5 \times 4\)[/tex]:
[tex]\[ 2 \times 5 = 10 \][/tex]
[tex]\[ 10 \times 4 = 40 \][/tex]
This expression equals 40, which is not equal to 1250.
2. [tex]\(2 \times 5 \times 5 \times 5 \times 5\)[/tex]:
Since we have already computed [tex]\(5^4 = 625\)[/tex], we can just multiply that by 2:
[tex]\[ 2 \times 625 = 1250 \][/tex]
This expression equals 1250, which matches [tex]\(2(5)^4\)[/tex].
3. [tex]\(2 \times 4 \times 4 \times 4 \times 4 \times 4\)[/tex]:
We need to calculate [tex]\(4^5\)[/tex] first:
[tex]\[ 4 \times 4 = 16 \][/tex]
[tex]\[ 16 \times 4 = 64 \][/tex]
[tex]\[ 64 \times 4 = 256 \][/tex]
[tex]\[ 256 \times 4 = 1024 \][/tex]
So, [tex]\(4^5 = 1024\)[/tex].
Now multiply by 2:
[tex]\[ 2 \times 1024 = 2048 \][/tex]
This expression equals 2048, which is not equal to 1250.
4. [tex]\(10 \times 10 \times 10 \times 10\)[/tex]:
[tex]\(10^4\)[/tex] means multiplying 10 by itself 4 times:
[tex]\[ 10 \times 10 = 100 \][/tex]
[tex]\[ 100 \times 10 = 1000 \][/tex]
[tex]\[ 1000 \times 10 = 10000 \][/tex]
This expression equals 10000, which is not equal to 1250.
From these evaluations, we see that the second expression [tex]\(2 \times 5 \times 5 \times 5 \times 5\)[/tex] is equivalent to [tex]\(2(5)^4\)[/tex].
Therefore, the equivalent expression to [tex]\(2(5)^4\)[/tex] is
[tex]\[ 2 \times 5 \times 5 \times 5 \times 5. \][/tex]
First, let's calculate [tex]\(2(5)^4\)[/tex]:
- [tex]\(5^4\)[/tex] means multiplying 5 by itself 4 times: [tex]\(5 \times 5 \times 5 \times 5\)[/tex].
- Now, let's calculate [tex]\(5 \times 5 \times 5 \times 5\)[/tex]:
[tex]\[ 5 \times 5 = 25 \][/tex]
[tex]\[ 25 \times 5 = 125 \][/tex]
[tex]\[ 125 \times 5 = 625 \][/tex]
- So, [tex]\(5^4 = 625\)[/tex].
- Now, multiply this result by 2:
[tex]\[ 2 \times 625 = 1250 \][/tex]
Thus, [tex]\(2(5)^4 = 1250\)[/tex].
Next, let's evaluate each of the given expressions and see which one equals 1250:
1. [tex]\(2 \times 5 \times 4\)[/tex]:
[tex]\[ 2 \times 5 = 10 \][/tex]
[tex]\[ 10 \times 4 = 40 \][/tex]
This expression equals 40, which is not equal to 1250.
2. [tex]\(2 \times 5 \times 5 \times 5 \times 5\)[/tex]:
Since we have already computed [tex]\(5^4 = 625\)[/tex], we can just multiply that by 2:
[tex]\[ 2 \times 625 = 1250 \][/tex]
This expression equals 1250, which matches [tex]\(2(5)^4\)[/tex].
3. [tex]\(2 \times 4 \times 4 \times 4 \times 4 \times 4\)[/tex]:
We need to calculate [tex]\(4^5\)[/tex] first:
[tex]\[ 4 \times 4 = 16 \][/tex]
[tex]\[ 16 \times 4 = 64 \][/tex]
[tex]\[ 64 \times 4 = 256 \][/tex]
[tex]\[ 256 \times 4 = 1024 \][/tex]
So, [tex]\(4^5 = 1024\)[/tex].
Now multiply by 2:
[tex]\[ 2 \times 1024 = 2048 \][/tex]
This expression equals 2048, which is not equal to 1250.
4. [tex]\(10 \times 10 \times 10 \times 10\)[/tex]:
[tex]\(10^4\)[/tex] means multiplying 10 by itself 4 times:
[tex]\[ 10 \times 10 = 100 \][/tex]
[tex]\[ 100 \times 10 = 1000 \][/tex]
[tex]\[ 1000 \times 10 = 10000 \][/tex]
This expression equals 10000, which is not equal to 1250.
From these evaluations, we see that the second expression [tex]\(2 \times 5 \times 5 \times 5 \times 5\)[/tex] is equivalent to [tex]\(2(5)^4\)[/tex].
Therefore, the equivalent expression to [tex]\(2(5)^4\)[/tex] is
[tex]\[ 2 \times 5 \times 5 \times 5 \times 5. \][/tex]
