Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{(5h^{-5}s^{-4})^{-3}}{20h^{5}s^{-7}}\\)

Explanation:

Step1: Apply the power of a product rule to the numerator.

The power of a product rule states that \((ab)^n = a^n b^n\). So, for \((5h^{-5}s^{-4})^{-3}\), we have:
\(5^{-3}(h^{-5})^{-3}(s^{-4})^{-3}\)
Using the power of a power rule \((a^m)^n = a^{mn}\), we get:
\(5^{-3}h^{(-5)\times(-3)}s^{(-4)\times(-3)} = 5^{-3}h^{15}s^{12}\)
So the numerator becomes \(5^{-3}h^{15}s^{12}\) and the denominator is \(20h^{5}s^{-7}\).

Step2: Simplify the coefficient and the variables separately.

First, simplify the coefficient:
\(\frac{5^{-3}}{20}\)
We know that \(5^{-3}=\frac{1}{5^{3}}=\frac{1}{125}\), so:
\(\frac{\frac{1}{125}}{20}=\frac{1}{125\times20}=\frac{1}{2500}\)
Next, simplify the variable \(h\):
Using the quotient rule for exponents \(\frac{a^m}{a^n}=a^{m - n}\), for \(h\) we have \(\frac{h^{15}}{h^{5}} = h^{15 - 5}=h^{10}\)
Then, simplify the variable \(s\):
Using the quotient rule for exponents \(\frac{a^m}{a^n}=a^{m - n}\), for \(s\) we have \(\frac{s^{12}}{s^{-7}} = s^{12-(-7)}=s^{12 + 7}=s^{19}\)

Step3: Combine the simplified coefficient and variables.

Multiply the simplified coefficient, \(h\) term, and \(s\) term together:
\(\frac{1}{2500}\times h^{10}\times s^{19}=\frac{h^{10}s^{19}}{2500}\)
We can also rewrite \(2500\) as \(5^{4}\) or \(2^{2}\times5^{4}\), but \(\frac{h^{10}s^{19}}{2500}\) is in simplest form with positive exponents.

Answer:

\(\frac{h^{10}s^{19}}{2500}\)