Question

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

Answer 1

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

### Step 1: Apply the Power of a Power Property
We start with \((5^{-3})^6\):
\[
(5^{-3})^6 = 5^{-3 \cdot 6} = 5^{-18}
\]

### Step 2: Substitute back into the original expression
Now our expression looks like this:
\[
5^{-18} \cdot \frac{5^{17}}{10^{-2}}
\]

### Step 3: Rewrite the fraction
Rewriting \(\frac{5^{17}}{10^{-2}}\), we can find that dividing by a negative exponent is equivalent to multiplying by a positive exponent:
\[
\frac{5^{17}}{10^{-2}} = 5^{17} \cdot 10^2
\]
Now we replace this in our expression to get:
\[
5^{-18} \cdot 5^{17} \cdot 10^2
\]

### Step 4: Combine the powers of \(5\)
Using the property \(a^m \cdot a^n = a^{m+n}\), we combine the powers of \(5\):
\[
5^{-18} \cdot 5^{17} = 5^{-18 + 17} = 5^{-1}
\]

So now we have:
\[
5^{-1} \cdot 10^2
\]

### Step 5: Rewrite \(5^{-1}\) with a positive exponent
The negative exponent can be rewritten as:
\[
5^{-1} = \frac{1}{5}
\]

Thus, our expression now is:
\[
\frac{1}{5} \cdot 10^2
\]

### Step 6: Calculate \(10^2\)
Calculating \(10^2\):
\[
10^2 = 100
\]

### Step 7: Simplify the entire expression
Now substituting back into our expression gives:
\[
\frac{1}{5} \cdot 100 = \frac{100}{5} = 20
\]

### Conclusion
The final result of the expression \((5^{-3})^6 \cdot \frac{5^{17}}{10^{-2}}\) with only positive exponents is:
\[
\boxed{20}
\]