Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (143⋅27−7)034
(
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)
(143⋅27−7)034=
(
14
3
⋅
27
−
7
)
0
3
4
=

Answer 1

no

Answer 2

To simplify the expression \((14^3 \cdot 27^{-7})^{034}\), we will first rewrite the expression using properties of exponents.

1. **Property of exponents: \(a^{m} \cdot a^{n} = a^{m+n}\)**
If values are negative, the expression should move to the denominator.

2. **Property of exponents: \((a \cdot b)^{n} = a^{n} \cdot b^{n}\)**
As we have a parenthesis raised to an exponent, we can distribute this exponent.

Starting with the original expression:
\[
(14^3 \cdot 27^{-7})^{034}
\]

We can rewrite this as:
\[
(14^3)^{034} \cdot (27^{-7})^{034}
\]

Using the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[
14^{3 \cdot 034} \cdot 27^{-7 \cdot 034}
\]

To simplify the multiplication:
\[
3 \cdot 034 = 102 \quad \text{and} \quad -7 \cdot 034 = -238
\]

Thus, we have:
\[
14^{102} \cdot 27^{-238}
\]

Now, we can rewrite \(27^{-238}\) as \(\frac{1}{27^{238}}\):
\[
\frac{14^{102}}{27^{238}}
\]

Next, we need to express \(27\) in terms of base \(3\):
\[
27 = 3^3
\]
So:
\[
27^{238} = (3^3)^{238} = 3^{3 \cdot 238} = 3^{714}
\]

Substituting back into our expression gives:
\[
\frac{14^{102}}{3^{714}}
\]

This is our final expression with no remaining exponents.

Thus, the solution is:
\[
\frac{14^{102}}{3^{714}}
\]

Answer 3

I apologize for the oversight. Let's carefully go through the process again step by step to simplify \((14^3 \cdot 27^{-7})^{034}\) into an equivalent expression with positive exponents and solve it.

### Step 1: Break Down the Expression
We have:
\[
(14^3 \cdot 27^{-7})^{034}
\]

### Step 2: Apply Exponent Properties
Using the property \((a \cdot b)^n = a^n \cdot b^n\), we can distribute the exponent:

\[
(14^3)^{034} \cdot (27^{-7})^{034}
\]

Next, applying the property \((a^m)^n = a^{m \cdot n}\):

\[
14^{3 \cdot 34} \cdot 27^{-7 \cdot 34}
\]

Calculating the exponents:
\[
3 \cdot 34 = 102 \quad \text{and} \quad -7 \cdot 34 = -238
\]

Thus, we rewrite the expression as:
\[
14^{102} \cdot 27^{-238}
\]

### Step 3: Convert \(27^{-238}\) to Positive Exponent
We can rewrite \(27^{-238}\) as:
\[
\frac{14^{102}}{27^{238}}
\]

### Step 4: Rewrite \(27\) Using Prime Factorization
Knowing that \(27\) can be rewritten in terms of \(3\):
\[
27 = 3^3 \quad \Rightarrow \quad 27^{238} = (3^3)^{238} = 3^{714}
\]

### Final Step: Write the Full Expression
Substituting this back into our expression gives:
\[
\frac{14^{102}}{3^{714}}
\]

This yields the final result:
\[
\frac{14^{102}}{3^{714}}
\]

### Conclusion
Given your prompt for a simplified fraction with no remaining exponents, the final result is:
\[
\frac{14^{102}}{3^{714}}
\]

If there’s a specific form you’d like for the answer or any adjustment needed, please let me know!