Simplify the following:
1) 32^1.2
2) (-216)^(-2/3)
3) -(-27)^(-4/3)
Note: 32^1.2 is 32 raised to the 1.2 power
1. 32=2^5
32^1.2=(2^5)^1.2=2^10
2. -216=-2*108=-2*2*54=-2*2*2*27=
=-2^3 *3^3
((-6)^3)^-2/3= Now the sign is tricky.
The cube root of a negative number is negative.
- 1/36
To simplify the given expressions, we can use the rules of exponentiation.
1) 32^1.2:
To simplify 32^1.2, we can rewrite it as (2^5)^1.2.
Using the rule for exponentiation of a power, we can multiply the exponents together, giving us 2^(5*1.2) = 2^6.
Therefore, 32^1.2 simplifies to 2^6, which equals 64.
2) (-216)^(-2/3):
To simplify (-216)^(-2/3), we can rewrite it as (-6^3)^(-2/3).
Using the rule (a^m)^n = a^(m*n), we can multiply the exponents together, giving us -6^(3*(-2/3)) = -6^-2.
Since -6^-2 is the reciprocal of -6^2, we can rewrite it as -1/(6^2).
Therefore, (-216)^(-2/3) simplifies to -1/36.
3) -(-27)^(-4/3):
To simplify -(-27)^(-4/3), we first simplify the exponent by multiplying -4/3 by the negative sign, giving us -4/3*(-1) = 4/3.
Now, we can rewrite the expression as -((-27)^(4/3)).
Using the rule -(-a) = a, we can further simplify it to (-27)^(4/3).
Since the exponent is positive in this case, we can calculate the value as (cube root of 27)^4 = 3^4 = 81.
Therefore, -(-27)^(-4/3) simplifies to 81.