1)5 [19+ (5^2-16) ^2] ^2
2)(7m^3-m-6m^2+10)-(5m^3-9+3m-2m^2)
3) 3 (a-2)(4a+1)-5(2a+3)(a-7)
4)6a^2b/-2ab^2
5) 120(1+i) ^2+180(1+i) ^3 /360(1+i)
1. 5[19+(5^2-16)^2]^2
5[19+(9)^2]^2
5[100]^2 = 50,000
2. (7m^3-6m^2+10)-(5m^3-9+3m-2m^2) =
7m^3-6m^2+10-5m^3+9-3m+2m^2 =
Combine like-terms:
2m^3-4m^2-3m+19
3. 3(a-2)(4a+1)-5(2a+3)(a-7) =
3(4a^2+1-8a-2)-5(2a^2-14a+3a-21) =
12a^2+3-24a-6-10a^2+70a-15a+105 =
Combine like-terms:
2a^2+31a+102.
4. 6a^2b/-2ab^2 = -3a^(b(2-b)).
5. 120(1+i)^2+180(1+i)^3/360(1+i) =
Divide the numerator and denominator of
the 2nd term by 180(1+i):
120(1+i)^2+(1+i)^2/2 =
(120 1/2)(1+i)^2 =
(241/2)(1+i)^2 =
(241/2)(1+2i+i^2) =
Replace i^2 with -1:
(241/2)(1+2i-1) =
(241/2)*2i = 241i.
(1+i)^2(120+180(1+i)/
NOTE: Disregard the last expression.
The answer is 241i.
1) To simplify the expression: 5 [19 + (5^2 - 16)^2]^2
First, let's solve the innermost exponent: (5^2 - 16) ^ 2
5^2 = 25
16 ^ 2 = 256
Substitute the values into the expression: (25 - 256)^2
Now, simplify the exponent: (-231)^2
(-231) ^ 2 = 53361
Next, substitute the simplified terms back into the original expression: 5 [19 + 53361]^2
Add the terms inside the square brackets: 19 + 53361 = 53380
Now, continue simplifying: 5 * (53380)^2
Next, calculate the square: 53380^2 = 2844366400
Finally, multiply by 5: 5 * 2844366400 = 14,222,832,000
Therefore, the simplified expression is 14,222,832,000.
2) To simplify the expression: (7m^3 - m - 6m^2 + 10) - (5m^3 - 9 + 3m - 2m^2)
Simply remove the parentheses without changing the signs of the terms inside:
7m^3 - m - 6m^2 + 10 - 5m^3 + 9 - 3m + 2m^2
Next, combine like terms. In this case, combine terms with the same degree of m:
(7m^3 - 5m^3) + (-6m^2 + 2m^2) + (-m - 3m) + (10 + 9)
Combine the like terms of m^3: 7m^3 - 5m^3 = 2m^3
Combine the like terms of m^2: -6m^2 + 2m^2 = -4m^2
Combine the like terms of m: -m - 3m = -4m
Combine the constants: 10 + 9 = 19
Therefore, the simplified expression is: 2m^3 - 4m^2 - 4m + 19.
3) To simplify the expression: 3(a-2)(4a+1) - 5(2a+3)(a-7)
Perform the multiplication using the distributive property:
3(a-2) = 3a - 6
4a + 1 = 4a + 1
5(2a + 3) = 10a + 15
(a-7) = a - 7
Now, substitute these simplified expressions back into the original equation:
3a - 6 * (4a + 1) - 10a - 15(a - 7)
Distribute the terms again:
3a - 6 * 4a - 6 * 1 - 10a - 15a + 15 * 7
Combine the like terms:
12a - 6 - 10a - 15a + 105
Combine the like terms:
12a - 10a - 15a - 6 + 105
Combine the like terms:
-13a + 99
Therefore, the simplified expression is -13a + 99.
4) To simplify the expression: 6a^2b / -2ab^2
First, divide the coefficients: 6a^2b / -2ab^2 = -3a^2b / ab^2
Next, simplify the variables: -3a^2b / ab^2 = -3a^(2-1)b^(1-2) = -3ab^(-1)
Therefore, the simplified expression is -3ab^(-1).
5) To simplify the expression: 120(1+i)^2 + 180(1+i)^3 / 360(1+i)
First, calculate the exponent expressions separately for each term:
(1+i)^2 = (1 + 2i + i^2) = 1 + 2i + (-1) = 2i
(1+i)^3 = (1 + 3i + 3i^2 + i^3) = 1 + 3i + 3(-1) + (-i) = -2 + 2i
Now substitute the results back into the original expression:
120 * 2i + 180 * (-2 + 2i) / 360(1+i)
Perform the multiplications:
240i - 360 + 360i / 360(1+i)
Combine the like terms:
(240i + 360i - 360) / 360(1+i)
Simplify the numerator:
600i - 360 / 360(1+i)
Now, divide the numerator and denominator separately:
600i / 360 = (10/6)i = (5/3)i
-360 / 360 = -1
So the simplified expression is:
(5/3)i - 1
Therefore, the simplified expression is (5/3)i - 1.