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: \[ \boxed{20} \]