Question

fully simplify using only positive exponents.\\(\frac{18x^{7}y^{5}}{30xy^{4}}\\)

Explanation:

Step1: Simplify the coefficients

First, simplify the fraction of the coefficients 18 and 30. Find the greatest common divisor (GCD) of 18 and 30, which is 6. Divide both 18 and 30 by 6:
$\frac{18}{30} = \frac{18\div6}{30\div6} = \frac{3}{5}$

Step2: Simplify the \(x\)-terms

For the \(x\)-terms, use the quotient rule of exponents: \( \frac{x^m}{x^n} = x^{m - n} \). Here, \( m = 7 \) and \( n = 1 \), so:
$\frac{x^7}{x} = x^{7 - 1} = x^6$

Step3: Simplify the \(y\)-terms

For the \(y\)-terms, use the quotient rule of exponents again. Here, \( m = 5 \) and \( n = 4 \), so:
$\frac{y^5}{y^4} = y^{5 - 4} = y^1 = y$

Step4: Combine the simplified terms

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term together:
$\frac{3}{5} \times x^6 \times y = \frac{3x^6y}{5}$

Answer:

$\frac{3x^6y}{5}$