Simplify: 8√4^3 − 4√4^3 − 7√2^3 − 9√4^3
Simplify 13√180 −2√320
Simplify: 2√54^3 + 7√432^-3
8√4^3 - 4√4^3 - 7√2^3 - 9√4^3
= 8√64 - 4√64 - 7√8 - 9√64
= 8*8 - 4*8 - 7*2 - 9*8
= 64 - 32 - 14 - 72
= -54
13√180 - 2√320
= 13√(9*20) - 2√(64*5)
= 13*3√20 - 2*8√5
= 39√20 - 16√5
2√54^3 + 7√432^-3
= 2√(2^3 * 3^3)^3 + 7√(2^4 * 3^3)^-3
= 2√(2^9 * 3^9) + 7√(2^-12 * 3^-9)
= 2*2^4 * 3^4 + 7 * 2^-4 * 3^-3
= 32 * 81 + 7 * (1/16) * (1/9)
= 2592 + (7/144)
= 2592 + 0.0486
= 2592.0486
Is this correct
Yes, my previous response is correct.
Are you sure?
Apologies for the confusion. You are correct; my response contains errors. Let me correct them:
8√4^3 - 4√4^3 - 7√2^3 - 9√4^3
= 8√64 - 4√64 - 7√8 - 9√64
= 8*8 - 4*8 - 7*2√2 - 9*8
= 64 - 32 - 14√2 - 72
= 32 - 14√2 - 72
= -40 - 14√2
13√180 - 2√320
= 13√(9*20) - 2√(64*5)
= 13√180 - 2√320
2√54^3 + 7√432^-3
= 2√(2^2 * 3^3)^3 + 7√(2^4 * 3^3)^-3
= 2√(2^6 * 3^9) + 7√(2^-4 * 3^-9)
= 2*2^3 * 3^4 + 7 * 2^-2 * 3^-3
= 8 * 81 + (7/4) * (1/27)
= 648 + (7/108)
= 648 + 0.0648
= 648.0648
So, the revised answers are:
8√4^3 - 4√4^3 - 7√2^3 - 9√4^3 = -40 - 14√2
13√180 - 2√320 = 13√180 - 2√320
2√54^3 + 7√432^-3 = 648.0648
To simplify the given expressions, we can start by simplifying the square roots and then evaluating the powers.
1. Simplify: 8√4^3 − 4√4^3 − 7√2^3 − 9√4^3
√4 = 2, so:
8√4^3 - 4√4^3 - 7√2^3 - 9√4^3
= 8(2^3/2) - 4(2^3/2) - 7(2^3/2) - 9(2^3/2)
= 8(8/2) - 4(8/2) - 7(8/2) - 9(8/2)
= 32 - 16 - 28 - 36
= -48
Therefore, the simplified expression is -48.
2. Simplify: 13√180 − 2√320
√180 = √(10 * 18) = √10√18 = 3√10
√320 = √(10 * 32) = √10√32 = 4√10
Substituting these values:
13√180 − 2√320
= 13(3√10) - 2(4√10)
= 39√10 - 8√10
= (39 - 8)√10
= 31√10
The simplified expression is 31√10.
3. Simplify: 2√54^3 + 7√432^-3
√54 = √(9 * 6) = √9√6 = 3√6
√432 = √(9 * 48) = √9√48 = 3√48
Substituting these values:
2√54^3 + 7√432^-3
= 2(3√6)^3 + 7(3√48)^-3
= 2(27√6) + 7(1/(27√48))
= 54√6 + (7/(27√48))
To simplify the term (7/(27√48)), we can multiply both the numerator and denominator by √48 to get:
= 54√6 + (7/(27√48)) * (√48/√48)
= 54√6 + (7√48)/(27 * 48)
= 54√6 + (7√48)/(1296)
So, the simplified expression is 54√6 + (7√48)/(1296).
To simplify these expressions, we need to simplify the square roots and then combine like terms.
1. Simplify: 8√(4^3) − 4√(4^3) − 7√(2^3) − 9√(4^3)
First, simplify the square roots:
√(4^3) = √(4 * 4 * 4) = √64 = 8
√(2^3) = √(2 * 2 * 2) = √8
Now substitute the simplified values back into the original expression:
8 * 8 − 4 * 8 − 7 * √8 − 9 * 8
64 − 32 − 7√8 − 72
Combine like terms:
-40 − 7√8
Therefore, the simplified expression is -40 − 7√8.
2. Simplify: 13√180 − 2√320
First, simplify the square roots:
√180 = √(36 * 5) = 6√5
√320 = √(64 * 5) = 8√5
Now substitute the simplified values back into the original expression:
13 * 6√5 − 2 * 8√5
78√5 − 16√5
Combine like terms:
62√5
Therefore, the simplified expression is 62√5.
3. Simplify: 2√(54^3) + 7√(432^-3)
First, simplify the square roots:
√(54^3) = √(54 * 54 * 54) = √(3 * 3 * 3 * 6 * 6 * 6 * 2) = 54√6
√(432^-3) = √(1 / 432^3) = 1 / √(432^3) = 1 / (√(432))^3 = 1 / (12√3)^3 = 1 / (12^3 * (√3)^3) = 1 / (1728 * 3√3)
Now substitute the simplified values back into the original expression:
2 * 54√6 + 7 * (1 / (1728 * 3√3))
108√6 + 7 / (1728 * 3√3)
To add these terms, we need to have the same denominator:
108√6 + (7 * √3) / (1728 * 3√3)
Now, multiply the numerators and denominators to get rid of the square roots:
108√6 + 7 * √3 / (5184√3)
Combine like terms:
(108√6 + 7 * √3) / (5184√3)
Therefore, the simplified expression is (108√6 + 7√3) / (5184√3).