If (9/7)3 × (49/81)(2x-6) = (7/9)9, then the value of x is:

Law of Exponents — Equal Bases Rule: if am = an for a base a with a ≠ 0, 1, -1, then the exponents must be equal, i.e. m = n. Two fractional bases that are reciprocals of each other, such as 9/7 and 7/9, can always be rewritten as powers of one another using (b/a) = (a/b)-1.

1. Rewrite every term with base 9/7. Since 49/81 = 72/92 = (7/9)2 = (9/7)-2, write (49/81)(2x-6) as (9/7)-2(2x-6).

2. Rewrite the right-hand side similarly: since 7/9 = (9/7)-1, (7/9)9 = (9/7)-9.

3. Substitute the rewritten terms so the whole equation has base 9/7 on both sides: (9/7)3 × (9/7)^(-2(2x-6)) = (9/7)-9.

4. Combine the exponents on the left-hand side using am·an = a(m+n): (9/7)3-2(2x-6) = (9/7)-9.

5. Apply the equal-bases rule to equate the exponents: 3 - 2(2x - 6) = -9.

6. Expand the bracket: 3 - 4x + 12 = -9, which simplifies to 15 - 4x = -9.

7. Solve for x: -4x = -9 - 15 = -24, so x = 6.

1. Substitute x = 6 back into the original equation: 2x - 6 = 6, so the middle term's exponent in base 9/7 is -2(6) = -12; adding the leading exponent 3 gives a total left-hand exponent of -9.

2. This matches the right-hand exponent -9 exactly (since 7/9 is the reciprocal of 9/7), so the equation balances at this value, confirming x = 6.