ARE YOU SURE YOU WANT TO SEE IT STEP BY STEP. IT WILL MAKE YOU DIZZY, BUT GOOD LUCK!!. HERE IT IS:
Simplify the following:
(-5 (-4))^3+((-6)^3)^2-(3^9/(-3)^8)^5
Simplify (-5 (-4))^3 by distributing exponents over products.
Multiply each exponent in -5 (-4) by 3:
(-5)^3×(-4)^3+((-6)^3)^2-(3^9/(-3)^8)^5
For all positive integer exponents (a^n)^m = a^(n m). Apply this to ((-6)^3)^2.
Multiply exponents. ((-6)^3)^2 = (-6)^(3×2):
(-5)^3×(-4)^3+(-6)^(3×2)-(3^9/(-3)^8)^5
Determine the sign of (-4)^3.
(-4)^3 = (-1)^3×4^3 = -4^3:
(-5)^3×-4^3+(-6)^(3×2)-(3^9/(-3)^8)^5
In order to evaluate 4^3 express 4^3 as 4×4^2.
4^3 = 4×4^2:
(-5)^3 (-4×4^2)+(-6)^(3×2)-(3^9/(-3)^8)^5
Evaluate 4^2.
4^2 = 16:
(-5)^3 (-4×16)+(-6)^(3×2)-(3^9/(-3)^8)^5
Multiply 4 and 16 together.
4×16 = 64:
(-5)^3 (-64)+(-6)^(3×2)-(3^9/(-3)^8)^5
Determine the sign of (-5)^3.
(-5)^3 = (-1)^3×5^3 = -5^3:
-5^3 (-64)+(-6)^(3×2)-(3^9/(-3)^8)^5
In order to evaluate 5^3 express 5^3 as 5×5^2.
5^3 = 5×5^2:
-5×5^2 (-64)+(-6)^(3×2)-(3^9/(-3)^8)^5
Evaluate 5^2.
5^2 = 25:
-5×25 (-64)+(-6)^(3×2)-(3^9/(-3)^8)^5
Multiply 5 and 25 together.
5×25 = 125:
-125 (-64)+(-6)^(3×2)-(3^9/(-3)^8)^5
Determine the sign of (-3)^8.
(-3)^8 = (-1)^8×3^8 = 1×3^8:
-125 (-64)+(-6)^(3×2)-(3^9/3^8)^5
Compute 3^8 by repeated squaring. For example a^7 = a a^6 = a (a^3)^2 = a (a a^2)^2.
3^8 = (3^4)^2 = ((3^2)^2)^2:
-125 (-64)+(-6)^(3×2)-(3^9/((3^2)^2)^2)^5
Evaluate 3^2.
3^2 = 9:
-125 (-64)+(-6)^(3×2)-(3^9/(9^2)^2)^5
Evaluate 9^2.
9^2 = 81:
-125 (-64)+(-6)^(3×2)-(3^9/81^2)^5
Evaluate 81^2.
| | 8 | 1
× | | 8 | 1
| | 8 | 1
6 | 4 | 8 | 0
6 | 5 | 6 | 1:
-125 (-64)+(-6)^(3×2)-(3^9/6561)^5
Compute 3^9 by repeated squaring. For example a^7 = a a^6 = a (a^3)^2 = a (a a^2)^2.
3^9 = 3×3^8 = 3 (3^4)^2 = 3 ((3^2)^2)^2:
-125 (-64)+(-6)^(3×2)-(3 ((3^2)^2)^2/6561)^5
Evaluate 3^2.
3^2 = 9:
-125 (-64)+(-6)^(3×2)-((3 (9^2)^2)/6561)^5
Evaluate 9^2.
9^2 = 81:
-125 (-64)+(-6)^(3×2)-((3×81^2)/6561)^5
Evaluate 81^2.
| | 8 | 1
× | | 8 | 1
| | 8 | 1
6 | 4 | 8 | 0
6 | 5 | 6 | 1:
-125 (-64)+(-6)^(3×2)-((3×6561)/6561)^5
Multiply 3 and 6561 together.
3×6561 = 19683:
-125 (-64)+(-6)^(3×2)-(19683/6561)^5
Reduce 19683/6561 to lowest terms. Start by finding the GCD of 19683 and 6561.
The gcd of 19683 and 6561 is 6561, so 19683/6561 = (6561×3)/(6561×1) = 6561/6561×3 = 3:
-125 (-64)+(-6)^(3×2)-3^5
Multiply 3 and 2 together.
3×2 = 6:
-125 (-64)+(-6)^6-3^5
Compute 3^5 by repeated squaring. For example a^7 = a a^6 = a (a^3)^2 = a (a a^2)^2.
3^5 = 3×3^4 = 3 (3^2)^2:
-125 (-64)+(-6)^6-3 (3^2)^2
Evaluate 3^2.
3^2 = 9:
-125 (-64)+(-6)^6-3×9^2
Evaluate 9^2.
9^2 = 81:
-125 (-64)+(-6)^6-3×81
Multiply 3 and 81 together.
3×81 = 243:
-125 (-64)+(-6)^6-243
Multiply -125 and -64 together.
| 1 | 2 | 5
× | | 6 | 4
| 5 | 0 | 0
7 | 5 | 0 | 0
8 | 0 | 0 | 0:
8000+(-6)^6-243
Determine the sign of (-6)^6.
(-6)^6 = (-1)^6×6^6 = 1×6^6:
8000+6^6-243
Compute 6^6 by repeated squaring. For example a^7 = a a^6 = a (a^3)^2 = a (a a^2)^2.
6^6 = (6^3)^2 = (6×6^2)^2:
8000+(6×6^2)^2-243
Evaluate 6^2.
6^2 = 36:
8000+(6×36)^2-243
Multiply 6 and 36 together.
6×36 = 216:
8000+216^2-243
Evaluate 216^2.
| | 2 | 1 | 6
× | | 2 | 1 | 6
| 1 | 2 | 9 | 6
| 2 | 1 | 6 | 0
4 | 3 | 2 | 0 | 0
4 | 6 | 6 | 5 | 6:
8000+46656-243
Evaluate 8000+46656 using long addition.
| 1 | | | |
| 4 | 6 | 6 | 5 | 6
+ | | 8 | 0 | 0 | 0
| 5 | 4 | 6 | 5 | 6:
54656-243
Subtract 243 from 54656.
| 5 | 4 | 6 | 5 | 6
- | | | 2 | 4 | 3
| 5 | 4 | 4 | 1 | 3:
Answer: |
| 54413
