Question

simplify the expression \\(\frac{15t^9}{(5t^4)^3}\\)

Explanation:

Step1: Simplify the denominator

First, we simplify the denominator \((5t^4)^3\) using the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n = a^{mn}\). So, \((5t^4)^3 = 5^3 \cdot (t^4)^3 = 125t^{12}\).

Step2: Simplify the fraction

Now we have the fraction \(\frac{15t^9}{125t^{12}}\). We can simplify the coefficients and the variables separately. For the coefficients, \(\frac{15}{125} = \frac{3}{25}\). For the variables, using the quotient rule for exponents \(\frac{a^m}{a^n} = a^{m - n}\), we have \(\frac{t^9}{t^{12}} = t^{9 - 12} = t^{-3} = \frac{1}{t^3}\) (since \(a^{-n}=\frac{1}{a^n}\)).

Step3: Combine the results

Multiplying the simplified coefficient and the simplified variable part together, we get \(\frac{3}{25} \cdot \frac{1}{t^3} = \frac{3}{25t^3}\).

Answer:

\(\frac{3}{25t^3}\)