use the Properties of Integer Exponents to generate equivalent expressions to 3^7⋅3^−9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the property that states:
a^m * a^n = a^(m + n).
Applying this property to the given expression, we have:
3^7 * 3^(-9) = 3^(7 + -9).
Exponents are the expressions that simplify to 0 can be removed, according to the property:
a^0 = 1 (except for a = 0).
Therefore:
3^(7 + -9) = 3^(-2).
To simplify a negative exponent, we use the property:
a^(-n) = 1 / a^n.
Applying this property to the expression 3^(-2), we get:
3^(-2) = 1 / 3^2.
Finally, solving 1 / 3^2:
1 / 3^2 = 1 / 9.
Thus, the expression 3^7 * 3^(-9) is equivalent to 1/9 when simplified with no remaining exponents.