simplify 9to 6th power times 3 to the 7th power all over 9 to the 8th power times 3 to the 6th power

I interpreted your question as

(9^6) (3^7)/(( 9^8)(3^6) )
= (1/9^2)(3)
= 3/81
= 1/27

To simplify the expression (9^6 * 3^7) / (9^8 * 3^6), we can use the rules of exponents.

First, let's simplify the numerator (9^6 * 3^7):

Using the exponent rule for multiplying powers with the same base, we add the exponents:
9^6 * 3^7 = 9^(6+7) * 3^7 = 9^13 * 3^7

Now, let's simplify the denominator (9^8 * 3^6):

Using the exponent rule for multiplying powers with the same base, we add the exponents:
9^8 * 3^6 = 9^(8+6) * 3^6 = 9^14 * 3^6

Now that we have simplified the numerator and the denominator, we can rewrite the expression:

(9^6 * 3^7) / (9^8 * 3^6) = (9^13 * 3^7) / (9^14 * 3^6)

Using the exponent rule for dividing powers with the same base, we subtract the exponents:
(9^13 * 3^7) / (9^14 * 3^6) = 9^(13-14) * 3^(7-6) = 9^(-1) * 3^1

Finally, we can simplify further using the rule that any number to the power of -1 is equal to its reciprocal:
9^(-1) * 3^1 = 1/9 * 3 = 3/9

So, the simplified expression is 3/9.