Question

Which of the following is an equivalent expression to
(2^-3 x 9^3)^-4/2^9 x 9^-10

Answer 1

To simplify this expression, we can use power rules and simplify each term separately before combining them.

Starting with (2^-3 x 9^3)^-4, we can simplify the brackets first.
2^-3 = 1/2^3 = 1/8
9^3 = 729
So, (2^-3 x 9^3)^-4 = (1/8 x 729)^-4 = (729/8)^-4

Moving on to 2^9 and 9^-10, let's simplify them.
2^9 = 512
9^-10 = 1/9^10 = 1/3486784401

So, the expression becomes:
(729/8)^-4 / 512 x 1/3486784401

To simplify further, we can also convert (729/8)^-4 to its reciprocal:
1 / (729/8)^4 / 512 x 1/3486784401

Now, let's calculate (729/8)^4:
(729/8)^4 = (531441/4096)^4

Finally, we have:
1 / (531441/4096)^4 / 512 x 1/3486784401

To simplify this expression even further, we can use properties of exponents to multiply the exponents in the denominator and divide the exponents in the numerator:
1 / [ (531441/4096)^4 * 512 ] / 3486784401

Now, we can simplify the expression:
1 / [ (531441/4096)^4 * 512 ] / 3486784401
= 1 / [ (531441/4096)^4 * 512 ] x 1/3486784401
= 1 / [ (531441/4096)^4 * 512 * 3486784401 ]

Therefore, the equivalent expression is 1 / [ (531441/4096)^4 * 512 * 3486784401 ].