Question

Apply the Properties of Integer Exponents to generate an equivalent expression to (5^-3)^6 times 5^17/10^-2 with only positive exponents, then solve the expression. Show your work.

Answer 1

the / means fraction

Answer 2

To solve the expression \((5^{-3})^6 \times \frac{5^{17}}{10^{-2}}\) using the properties of integer exponents, we will follow these steps:

1. **Apply the exponent rule** \((a^m)^n = a^{mn}\) to simplify \((5^{-3})^6\).
2. **Simplify the fraction** \(\frac{5^{17}}{10^{-2}}\).
3. **Combine the results and simplify further**.

### Step 1: Simplify \((5^{-3})^6\)

Using the exponent rule \((a^m)^n = a^{mn}\):

\[
(5^{-3})^6 = 5^{-18}
\]

### Step 2: Simplify \(\frac{5^{17}}{10^{-2}}\)

Recall that \(10^{n} = 2^{n} \cdot 5^{n}\). Therefore, we can rewrite \(10^{-2}\) as:

\[
10^{-2} = \frac{1}{10^2} = \frac{1}{2^2 \cdot 5^2} = \frac{1}{4 \cdot 25} = \frac{1}{100}
\]

Hence,

\[
\frac{5^{17}}{10^{-2}} = 5^{17} \cdot 10^2 = 5^{17} \cdot 100 = 5^{17} \cdot (10^2) = 5^{17} \cdot (2^2 \cdot 5^2) = 5^{17} \cdot 2^2 \cdot 5^2
\]

This further simplifies to:

\[
5^{17} \cdot 4 \cdot 5^2 = 4 \cdot 5^{17 + 2} = 4 \cdot 5^{19}
\]

### Step 3: Combine Results

Now we can combine \(5^{-18}\) and \(4 \cdot 5^{19}\):

\[
5^{-18} \cdot (4 \cdot 5^{19}) = 4 \cdot 5^{-18 + 19} = 4 \cdot 5^{1}
\]

### Step 4: Rewrite with Positive Exponents

Since \(5^1 = 5\), we have:

\[
4 \cdot 5 = 20
\]

Thus, the final equivalent expression with only positive exponents is:

\[
\boxed{20}
\]

Answer 3

Thank you for clarifying. Let's resolve the expression correctly, accounting for the division properly. The expression we need to simplify is:

\[
(5^{-3})^6 \times \frac{5^{17}}{10^{-2}}
\]

### Step 1: Simplify \((5^{-3})^6\)

Using the exponent rule \((a^m)^n = a^{mn}\):

\[
(5^{-3})^6 = 5^{-18}
\]

### Step 2: Rewrite \(\frac{5^{17}}{10^{-2}}\)

We can rewrite \(10^{-2}\) using its prime factorization:

\[
10 = 2 \times 5 \quad \Rightarrow \quad 10^{-2} = (2 \times 5)^{-2} = 2^{-2} \cdot 5^{-2}
\]

Thus, we can rewrite the fraction:

\[
\frac{5^{17}}{10^{-2}} = 5^{17} \cdot 10^2 = 5^{17} \cdot (2^2 \cdot 5^2)
\]

This results in:

\[
\frac{5^{17}}{10^{-2}} = 5^{17} \cdot 2^2 \cdot 5^2 = 4 \cdot 5^{17 + 2} = 4 \cdot 5^{19}
\]

### Step 3: Combine the Results

Now we can combine \(5^{-18}\) and \(4 \cdot 5^{19}\):

\[
5^{-18} \cdot (4 \cdot 5^{19}) = 4 \cdot 5^{-18 + 19} = 4 \cdot 5^1
\]

### Step 4: Final Calculation

Thus, we have:

\[
4 \cdot 5 = 20
\]

Therefore, the final simplified expression is:

\[
\boxed{20}
\]