Question

simplify.\\(
\frac{12(x - 7)^{5}(2x - 3)^{3}}{40(2x - 3)^{4}(x - 7)^{4}}
\\)you may leave the numerator and denominator of your answer in factored form.

Explanation:

Step1: Simplify the numerical coefficients

First, simplify the fraction of the numerical coefficients \( \frac{12}{40} \). We can divide both the numerator and the denominator by their greatest common divisor, which is 4. So, \( \frac{12\div4}{40\div4}=\frac{3}{10} \).

Step2: Simplify the \((x - 7)\) terms

For the \((x - 7)\) terms, we use the rule of exponents \( \frac{a^m}{a^n}=a^{m - n} \). Here, \( m = 5 \) and \( n = 4 \), so \( \frac{(x - 7)^5}{(x - 7)^4}=(x - 7)^{5-4}=x - 7 \).

Step3: Simplify the \((2x - 3)\) terms

For the \((2x - 3)\) terms, again using the rule of exponents \( \frac{a^m}{a^n}=a^{m - n} \). Here, \( m = 3 \) and \( n = 4 \), so \( \frac{(2x - 3)^3}{(2x - 3)^4}=(2x - 3)^{3-4}=(2x - 3)^{-1}=\frac{1}{2x - 3} \).

Step4: Combine all the simplified terms

Now, multiply all the simplified parts together: \( \frac{3}{10}\times(x - 7)\times\frac{1}{2x - 3}=\frac{3(x - 7)}{10(2x - 3)} \).

Answer:

\( \dfrac{3(x - 7)}{10(2x - 3)} \)