Question

Which of the following is an equivalent expression to
(2^-3 x 9^3)^-4/2^9 x 9^-10 With only positive exponents, generated by applying the properties of integer exponents

Answer 1

To simplify the expression and generate only positive exponents, we can apply the properties of integer exponents.

First, let's simplify each term individually:

(2^-3 x 9^3) = (1/2^3 x 9^3) = 1/8 x 729 = 729/8

(2^9 x 9^-10) = (2^9 x 1/9^10) = 512/9^10

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

(729/8)^-4/512/9^10

Using the properties of exponents, when we raise a fraction to a negative power, we invert the fraction and make the exponent positive:

(8/729)^4/512/9^10

Now, let's simplify the expression inside the parentheses separately:

(8/729)^4 = 8^4/729^4 = 4096/531441
(512/9^10) = 512/9^10

Now, substitute these values back into the expression:

(4096/531441)/(512/9^10)

To divide fractions, we multiply by the reciprocal of the second fraction:

(4096/531441) x (9^10/512)

Now, let's simplify each term individually again:

(4096/531441) = 8/729
(9^10) = 3486784401
(512) = 2^9 = 512

Substitute these values back into the expression:

(8/729) x (3486784401/512)

To multiply fractions, we simply multiply the numerators together and the denominators together:

(8 x 3486784401)/(729 x 512)

Finally, simplify the result:

27894275208/373248 = 746496/1 = 746496

Therefore, the equivalent expression with only positive exponents is 746496.