simplify the following and express Answer in positive index
(e^3/f^2)^-3 x (e^-4/f)^-1
8^1/3 X 4^1/2
(e^3/f^2)^-3 * (e^-4/f)^-1
= (f^2/e^3)^3 * (f/e^-4)
= f^6/e^9 * e^4f
= f^7/e^5
8^(1/3) * 4^(1/2)
since 8=2^3 and 4=2^2, that's just
= 2 * 2
= 4
(e ^ 3 / f ^ 2 ) ^ - 3 = 1 / (e ^ 3 / f ^ 2 ) ^ 3 = 1 / ( e ^ 9 / f ^ 6 ) = f ^ 6 / e ^ 9
( e ^ - 4 / f ) ^ - 1 = 1 / [ ( 1 / e ^ 4 ) / f ] = 1 / [ 1 / ( e ^ 4 * f ) ] = e ^ 4 * f / 1 = e ^ 4 * f
(e ^ 3 / f ^ 2 ) ^ - 3 * ( e ^ - 4 / f ) ^ - 1 = ( f ^ 6 / e ^ 9 ) * e ^ 4 * f = f ^ 6 * f * e ^ 4 / e ^ 9 = f ^ 7 / e ^ 5 = ( f ^ 5 / e ^ 5 ) * f ^ 2 = ( f / e ) ^ 5 * f ^ 2
8 ^ ( 1 / 3 ) = third root ( 8 )
4 ^ ( 1 / 2 ) = sqroot ( 4 )
8 ^ ( 1 / 3 ) * 4 ^ ( 1 / 2 ) = third root ( 8 ) * sqroot ( 4 ) = 2 * 2 = 4
To simplify the given expressions and express the answers in positive index form, we will use the properties of exponentiation.
1. Simplifying (e^3/f^2)^-3 x (e^-4/f)^-1:
First, we can rewrite the expression using the property that raising a power to a negative exponent is the same as taking the reciprocal of the base and changing the sign of the exponent. Additionally, multiplying two powers with the same base is equivalent to adding their exponents. Applying these rules, we have:
(e^3/f^2)^-3 x (e^-4/f)^-1
= (f^2/e^3)^3 x (f/e^-4)
= f^(2*3)/e^(3*3) x f/e^4
= f^6/e^9 x f/e^4
Now, to multiply these two fractions, we combine the numerators and the denominators respectively:
(f^6)(f) / (e^9)(e^4)
= f^7 / e^13
Therefore, the simplified expression in positive index form is f^7 / e^13.
2. Simplifying 8^1/3 x 4^1/2:
To simplify this expression, let's rewrite the numbers as powers. Recall that taking the nth root of a number is the same as raising it to the 1/n power. We have:
8^1/3 x 4^1/2
= (2^3)^1/3 x (2^2)^1/2
= 2^(3*(1/3)) x 2^(2*(1/2))
= 2^(1) x 2^(1)
= 2 x 2
= 4
Hence, the simplified expression in positive index form is 4.