Question

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

Answer 1

To simplify the expression \((143 \cdot 27^{-7})^{034}\), we can start by applying the properties of integer exponents.

### Step 1: Simplify the exponent

The exponent \(034\) is simply \(34\), so we rewrite:

\[
(143 \cdot 27^{-7})^{34}
\]

### Step 2: Apply the power of a product property

Using the power of a product property, we can distribute the exponent:

\[
(143)^{34} \cdot (27^{-7})^{34}
\]

### Step 3: Apply the power of a power property

Now, applying the power of a power property to the second term:

\[
(27^{-7})^{34} = 27^{-7 \cdot 34} = 27^{-238}
\]

So our expression becomes:

\[
143^{34} \cdot 27^{-238}
\]

### Step 4: Convert to positive exponents

We can rewrite \(27^{-238}\) as:

\[
\frac{143^{34}}{27^{238}}
\]

Now, we have:

\[
(143 \cdot 27^{-7})^{034} = \frac{143^{34}}{27^{238}}
\]

### Step 5: Apply Exponent to Base 27

Next, we rewrite the base \(27\) using its prime factorization. Since \(27 = 3^3\), we can express it as:

\[
27^{238} = (3^3)^{238} = 3^{3 \cdot 238} = 3^{714}
\]

Thus, the expression becomes:

\[
\frac{143^{34}}{3^{714}}
\]

### Final result:

The simplified form of the original expression in terms of positive exponents is:

\[
\frac{143^{34}}{3^{714}}
\]

As a result, the expression without remaining exponents is:

\[
\frac{143^{34}}{3^{714}}
\]

This is your final simplified expression.