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.
To apply the properties of integer exponents to generate equivalent expressions with only positive exponents, we can simplify the given expression step by step:
Step 1: Rewrite the expression using the property a^n / b^n = (a/b)^n.
[(14^3)/(27^-7)]^(0/3^4)
Step 2: Apply the property (a^m)^n = a^(m*n) to the numerator.
[(14^(3*1))/(27^(-7*1))]^(0/3^4)
Step 3: Simplify the exponents in the numerator and denominator.
(14^3 / 27^-7)^(0/3^4)
Step 4: Apply the property (a^-n) = (1/a^n) to move the negative exponent in the denominator.
(14^3 / (1/27^7))^(0/3^4)
Step 5: Simplify the expression in the denominator.
(14^3 / 27^7)^(0/3^4)
Step 6: Apply the property a^0 = 1 to eliminate the exponent of zero.
1
Therefore, the equivalent expression with only positive exponents simplifies to 1.
So, the solution to the expression is 1.
wrong
Apologies for the mistake. Let's try again:
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can simplify the given expression step by step:
Step 1: Simplify the exponents in the numerator and denominator.
(14^3 - 27^(-7))^(0/3^4)
Step 2: Solve the powers with negative exponents by applying the property a^(-n) = 1/a^n.
(14^3 - 1/27^7)^(0/3^4)
Step 3: Apply the property a^0 = 1 to eliminate the exponent of zero.
(14^3 - 1/27^7)^0
Step 4: Any expression raised to the power of zero simplifies to 1.
1
Therefore, the equivalent expression with only positive exponents simplifies to 1.
So, the solution to the expression is 1.
To generate equivalent expressions with only positive exponents, we can rewrite the expression using the properties of integer exponents.
The first property states that (a^m)^n = a^(m * n). Using this property, we can simplify the expression:
(14^3 - 27^(-7))^0 / 3^4
= (14^(3*0) - 27^(-7*0)) / 3^4
= (14^0 - 27^0) / 3^4
= (1 - 1) / 3^4
= 0 / 3^4
= 0
Hence, the expression simplifies to 0.