Question
Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27^−7)^0/3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents. (1 point)
(14^3⋅27^−7)^0/3^4 = ?
(14^3⋅27^−7)^0/3^4 = ?
Answers
Answered by
GPT 3.5
To apply the properties of integer exponents, we can use the following rules:
1) For any non-zero number a, a^0 = 1.
2) For any non-zero number a, a^m/n = (a^m)^(1/n) = nth root of (a^m).
Let's start by expanding the expression (14^3⋅27^−7)^0:
(14^3⋅27^−7)^0 = 14^0 ⋅ (27^−7)^0.
Using the first rule, 14^0 = 1, and since anything raised to the power of 0 equals 1, we can simplify the expression further:
1 ⋅ (27^−7)^0 = 1 ⋅ 1 = 1.
Now let's focus on the denominator, 3^4. We can apply rule number 2, which states that (a^m/n) = (a^m)^(1/n). In our case, a = 3, m = 4, and n = 1, so we have:
3^4 = (3^4)^(1/1) = 3^(4/1) = 3^4.
Now we can substitute the values back into the original expression and solve it:
(14^3⋅27^−7)^0/3^4 = 1/3^4.
Since 3^4 = 81, the final expression can be simplified to:
1/3^4 = 1/81.
Therefore, the equivalent expression with only positive exponents is 1/81.
1) For any non-zero number a, a^0 = 1.
2) For any non-zero number a, a^m/n = (a^m)^(1/n) = nth root of (a^m).
Let's start by expanding the expression (14^3⋅27^−7)^0:
(14^3⋅27^−7)^0 = 14^0 ⋅ (27^−7)^0.
Using the first rule, 14^0 = 1, and since anything raised to the power of 0 equals 1, we can simplify the expression further:
1 ⋅ (27^−7)^0 = 1 ⋅ 1 = 1.
Now let's focus on the denominator, 3^4. We can apply rule number 2, which states that (a^m/n) = (a^m)^(1/n). In our case, a = 3, m = 4, and n = 1, so we have:
3^4 = (3^4)^(1/1) = 3^(4/1) = 3^4.
Now we can substitute the values back into the original expression and solve it:
(14^3⋅27^−7)^0/3^4 = 1/3^4.
Since 3^4 = 81, the final expression can be simplified to:
1/3^4 = 1/81.
Therefore, the equivalent expression with only positive exponents is 1/81.