Answer :
Certainly! Let's break down the expression step by step using the order of operations:
1. Evaluate the first expression:
[tex]\[
\left[\left(3568 - \left(13^2 - 5^2\right) \cdot 23\right) : 64\right]^2
\][/tex]
a. Calculate [tex]\(13^2\)[/tex] and [tex]\(5^2\)[/tex]:
- [tex]\(13^2 = 169\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
b. Subtract these squares:
- [tex]\(169 - 25 = 144\)[/tex]
c. Multiply the result by 23:
- [tex]\(144 \times 23 = 3312\)[/tex]
d. Subtract from 3568:
- [tex]\(3568 - 3312 = 256\)[/tex]
e. Divide by 64:
- [tex]\(256 / 64 = 4\)[/tex]
f. Square the result:
- [tex]\(4^2 = 16\)[/tex]
2. Evaluate the second expression:
[tex]\[
\left[1050 - 1000 : (100 : 5)\right] : 8 - \left(164 - 4 - 6^2\right)^{100}
\][/tex]
a. Calculate [tex]\(100 : 5\)[/tex]:
- [tex]\(100 / 5 = 20\)[/tex]
b. Divide 1000 by this result:
- [tex]\(1000 / 20 = 50\)[/tex]
c. Subtract from 1050:
- [tex]\(1050 - 50 = 1000\)[/tex]
d. Divide by 8:
- [tex]\(1000 / 8 = 125\)[/tex]
e. Solve [tex]\(164 - 4 - 6^2\)[/tex]:
- Calculate [tex]\(6^2 = 36\)[/tex]
- [tex]\(164 - 4 - 36 = 124\)[/tex]
f. Use this to raise 100 to the power of the result:
- [tex]\(100^{124}\)[/tex]
3. Evaluate the third expression:
[tex]\[
\left\{3 + \left(6 + 2^2\right) \cdot \left[2 \cdot 181 + 10 \cdot \left(24 + 24 : 2^2\right)\right]\right\}
\][/tex]
a. Calculate [tex]\(2^2\)[/tex]:
- [tex]\(2^2 = 4\)[/tex]
b. Add 6:
- [tex]\(6 + 4 = 10\)[/tex]
c. Solve [tex]\(24 + 24 : 2^2\)[/tex]:
- Calculate [tex]\(24 / 4 = 6\)[/tex]
- [tex]\(24 + 6 = 30\)[/tex]
d. Multiply 10 by this result:
- [tex]\(10 \times 30 = 300\)[/tex]
e. Multiply 2 by 181:
- [tex]\(2 \times 181 = 362\)[/tex]
f. Add the two multiplication results:
- [tex]\(362 + 300 = 662\)[/tex]
g. Multiply by the 10 calculated before:
- [tex]\(10 \times 662 = 6620\)[/tex]
h. Finally, add 3 to this:
- [tex]\(3 + 6620 = 6623\)[/tex]
The numerical results for the three expressions are:
- First expression: [tex]\(16\)[/tex]
- Second expression: A very large number (as calculated, [tex]\(100^{124}\)[/tex])
- Third expression: [tex]\(6623\)[/tex]
These results illustrate the step-by-step resolutions for each expression, following the order of operations precisely.
1. Evaluate the first expression:
[tex]\[
\left[\left(3568 - \left(13^2 - 5^2\right) \cdot 23\right) : 64\right]^2
\][/tex]
a. Calculate [tex]\(13^2\)[/tex] and [tex]\(5^2\)[/tex]:
- [tex]\(13^2 = 169\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
b. Subtract these squares:
- [tex]\(169 - 25 = 144\)[/tex]
c. Multiply the result by 23:
- [tex]\(144 \times 23 = 3312\)[/tex]
d. Subtract from 3568:
- [tex]\(3568 - 3312 = 256\)[/tex]
e. Divide by 64:
- [tex]\(256 / 64 = 4\)[/tex]
f. Square the result:
- [tex]\(4^2 = 16\)[/tex]
2. Evaluate the second expression:
[tex]\[
\left[1050 - 1000 : (100 : 5)\right] : 8 - \left(164 - 4 - 6^2\right)^{100}
\][/tex]
a. Calculate [tex]\(100 : 5\)[/tex]:
- [tex]\(100 / 5 = 20\)[/tex]
b. Divide 1000 by this result:
- [tex]\(1000 / 20 = 50\)[/tex]
c. Subtract from 1050:
- [tex]\(1050 - 50 = 1000\)[/tex]
d. Divide by 8:
- [tex]\(1000 / 8 = 125\)[/tex]
e. Solve [tex]\(164 - 4 - 6^2\)[/tex]:
- Calculate [tex]\(6^2 = 36\)[/tex]
- [tex]\(164 - 4 - 36 = 124\)[/tex]
f. Use this to raise 100 to the power of the result:
- [tex]\(100^{124}\)[/tex]
3. Evaluate the third expression:
[tex]\[
\left\{3 + \left(6 + 2^2\right) \cdot \left[2 \cdot 181 + 10 \cdot \left(24 + 24 : 2^2\right)\right]\right\}
\][/tex]
a. Calculate [tex]\(2^2\)[/tex]:
- [tex]\(2^2 = 4\)[/tex]
b. Add 6:
- [tex]\(6 + 4 = 10\)[/tex]
c. Solve [tex]\(24 + 24 : 2^2\)[/tex]:
- Calculate [tex]\(24 / 4 = 6\)[/tex]
- [tex]\(24 + 6 = 30\)[/tex]
d. Multiply 10 by this result:
- [tex]\(10 \times 30 = 300\)[/tex]
e. Multiply 2 by 181:
- [tex]\(2 \times 181 = 362\)[/tex]
f. Add the two multiplication results:
- [tex]\(362 + 300 = 662\)[/tex]
g. Multiply by the 10 calculated before:
- [tex]\(10 \times 662 = 6620\)[/tex]
h. Finally, add 3 to this:
- [tex]\(3 + 6620 = 6623\)[/tex]
The numerical results for the three expressions are:
- First expression: [tex]\(16\)[/tex]
- Second expression: A very large number (as calculated, [tex]\(100^{124}\)[/tex])
- Third expression: [tex]\(6623\)[/tex]
These results illustrate the step-by-step resolutions for each expression, following the order of operations precisely.
