Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27^−7)^0\3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(14^3⋅27^−7)^0\3^4=

Answer 1

To apply the properties of integer exponents, we can rewrite the expression as:

(14^3 * 27^(-7))^(1/3)^4

Now, let's simplify each term separately:

14^3 = 14 * 14 * 14 = 2,744

27^(-7) = 1/(27^7)

Now, let's simplify the expression further:

(2,744 * (1/(27^7)))^(1/3)^4

(2,744 * (1/823543))^(1/3)^4

Now, let's simplify the term in brackets:

(2,744/823543)^(1/3)^4

Now, let's simplify the exponent:

(2,744/823543)^(4/3)

To simplify the expression further, we can take the cube root of both the numerator and the denominator:

(14/27)^(4/3)

To simplify the exponent further, we can expand it as a fractional exponent:

(14/27)^(4/3) = ((14)^(4/3))/(27^(4/3))

Now, let's simplify each term separately:

(14)^(4/3) = (14^4)^(1/3) = 38416^(1/3) = 46

(27)^(4/3) = (27^4)^(1/3) = 531441^(1/3) = 81

Now, let's substitute these values back into the expression:

((14)^(4/3))/(27^(4/3)) = 46/81

Therefore, the simplified fraction is 46/81.