apply the properties of integer exponents to generate equivalent expressions to (14^3x27^-7)^0/3^4 with only positive exponents, then solve the expression. your answer will be a simplified fraction with no remaining exponents.
To apply the properties of integer exponents, we can simplify the expression as follows:
(14^3 × 27^(-7))/(3^4)
Now, let's simplify each component individually:
14^3 = (2 × 7)^3 = 2^3 × 7^3 = 8 × 343 = 2744
27^(-7) = 1/27^7 = 1/((3^3)^7) = 1/3^(3 × 7) = 1/3^21
3^4 = 3 × 3 × 3 × 3 = 81
Substituting these values back into the expression:
(2744 × (1/3^21))/81
Now, let's simplify further:
2744 × (1/3^21) = 2744/3^21
To simplify the fraction, we can find common factors in the numerator and denominator. In this case, both 2744 and 3^21 have a factor of 3:
2744 = 2 × 2 × 2 × 7 × 7 × 7 = 8 × 343 = 8 × (3^3 × 7^3) = 8 × 3^3 × 7^3
Thus, we can rewrite 2744 as 8 × 3^3 × 7^3. Now our expression becomes:
(8 × 3^3 × 7^3)/(3^21)/81
Next, we can simplify further by canceling out common factors:
(8 × 7^3)/(3^21)/(81/1)
The fraction 81/1 can be rewritten as 1/81, resulting in:
(8 × 7^3)/(3^21) × 1/81
Finally, we can simplify the expression by multiplying numerators and denominators:
(8 × 7^3)/(3^21 × 81)
= (8 × 343)/(3^21 × 81)
= (2744)/(3^21 × 81)
Hence, the simplified fraction with no remaining exponents is 2744/(3^21 × 81).