Apply the Properties of Integer Exponents to generate equivalent expressions to 37⋅3−9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
To generate an expression with only positive exponents for 37⋅3−9, we can use the property of negative exponents which states that a^(-n) = 1/a^n.
So, using this property, we have:
37⋅3^(-9) = 37/(3^9).
Now, let's simplify 3^9.
3^9 = 3^(3 * 3) = (3^3)^3 = 27^3 = 19683.
So, the equivalent expression with only positive exponents is 37/19683.
Therefore, the solved expression is 37/19683.
To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:
1. Product of Powers Property: a^n ⋅ a^m = a^(n+m)
2. Power of a Power Property: (a^n)^m = a^(n⋅m)
3. Power of a Product Property: (ab)^n = a^n ⋅ b^n
Starting with the expression 37⋅3^(-9), we can apply the properties as follows:
1. Rewrite 3^(-9) as the reciprocal of 3^9: 37 ⋅ (1/(3^9))
2. Apply the Power of a Product Property: 37 ⋅ (1^9)/(3^9)
3. Apply the Power of a Power Property: 37 ⋅ 1^9/(3^9)
4. Simplify 1^9 to just 1: 37/(3^9)
Therefore, the equivalent expression with only positive exponents is 37/(3^9).
To solve this expression, we need to simplify it further:
3^9 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 19683
Thus, 37/(3^9) = 37/19683, which is the simplified fraction with no remaining exponents.