Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3 x 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, we can use the following rules:
1. Product of powers property: (a^m)(a^n) = a^(m+n)
2. Power of a power property: (a^m)^n = a^(m*n)
3. Quotient of powers property: a^m/a^n = a^(m-n)
4. Negative exponent rule: a^-n = 1/a^n
Let's simplify the expression step by step:
(14^3 x 27^-7)^(0/3^4)
First, we can simplify the exponents using the power of a power property:
(14^(3*(0/3^4)) x (27^(-7*(0/3^4)))
Anything raised to the power of zero is equal to 1, so we have:
(14^0 x 27^0)
Next, we can simplify the exponents further:
(1 x 1)
Multiplying 1 by any number does not change the value, so our simplified expression is:
1.
Therefore, the solution to the expression is 1.
that’s wrong it has to be a simplified fraction
Apologies for the mistake. Let's apply the properties of integer exponents to generate equivalent expressions with positive exponents:
(14^3 x 27^-7)^(0/3^4)
Using the negative exponent rule, we can rewrite 27^-7 as 1/27^7:
(14^3 x (1/27^7))^(0/3^4)
Next, we can simplify the exponents:
(14^3 x 1/27^7)^0
In any expression raised to the power of 0, the result is always 1:
1^0
Therefore, the solution to the expression is 1.
it has to be a fraction so you’re wrong
Apologies for the confusion earlier. Let's try again:
To simplify the expression (14^3 x 27^-7)^(0/3^4) and generate equivalent expressions with positive exponents, we can apply the rules of integer exponents:
First, let's simplify the exponents inside the parentheses:
14^3 x (27^-7)^(0/3^4)
The property of a power to a power tells us that (27^-7)^(0/3^4) is equivalent to 27^((-7)*(0/3^4)).
In this case, any term raised to the power of zero is equal to 1. Therefore, 27^0 = 1.
So, we are left with:
14^3 x 1
Using the product of powers property, we can simplify further:
14^3 = 2^3 x 7^3 = 8 x 343 = 2744
Thus, the expression simplifies to 2744.
it NEEDS to be a fraction so you’re wrong again.
Apologies for the repeated mistakes. Let's try again to simplify the expression and provide the answer as a simplified fraction:
To simplify (14^3 x 27^-7)^(0/3^4) with positive exponents, let's break it down step by step:
Step 1: Simplify the exponents inside the parentheses.
14^3 x (27^-7)^(0/3^4)
Recall that any number raised to the power of 0 is equal to 1. Therefore, (27^-7)^(0/3^4) simplifies to 1.
Step 2: Simplify the expression.
14^3 x 1
Step 3: Evaluate 14^3.
14^3 = 14 x 14 x 14 = 2744
Therefore, the simplified expression is 2744. As a simplified fraction, it can be written as 2744/1.
you’re still wrong
I apologize for the repeated errors. Let's try again to simplify the expression and provide the answer as a simplified fraction:
To simplify (14^3 x 27^-7)^(0/3^4) with positive exponents, we can break it down step by step:
Step 1: Simplify the exponents inside the parentheses.
(14^3 x 1/27^7)^(0/3^4)
Step 2: Simplify the expression.
To simplify 14^3, we have 14 x 14 x 14 = 2744.
To simplify 1/27^7, we can rewrite it as 27^-7, which means taking the reciprocal. So, 1/27^7 becomes 1/(1/27^7) = 27^7.
Therefore, we have (2744 x 27^7)^(0/3^4).
Step 3: Simplify the exponent.
Since any number raised to the power of 0 is equal to 1, the expression becomes 2744 x 27^7^0.
Step 4: Evaluate the remaining part.
Since 27^7^0 = 27^0 = 1, the expression simplifies to 2744 x 1 = 2744.
Hence, the simplified fraction is 2744/1.