Question

fully simplify using only positive exponents. \\(\frac{18x^{7}y^{7}}{30x^{5}y^{6}}\\)

Explanation:

Step1: Simplify the coefficients

First, simplify the fraction of the coefficients 18 and 30. The greatest common divisor of 18 and 30 is 6. So, we divide both 18 and 30 by 6:
$\frac{18}{30} = \frac{18\div6}{30\div6} = \frac{3}{5}$

Step2: Simplify the \(x\)-terms using the quotient rule of exponents

The quotient rule for exponents states that $\frac{a^m}{a^n} = a^{m - n}$ (where \(a
eq0\), \(m\) and \(n\) are real numbers). For the \(x\)-terms, we have $\frac{x^7}{x^5}$. Applying the quotient rule:
$\frac{x^7}{x^5} = x^{7 - 5} = x^2$

Step3: Simplify the \(y\)-terms using the quotient rule of exponents

For the \(y\)-terms, we have $\frac{y^7}{y^6}$. Applying the quotient rule of exponents:
$\frac{y^7}{y^6} = y^{7 - 6} = y^1 = y$

Step4: Combine the simplified parts

Now, we combine the simplified coefficient, \(x\)-term, and \(y\)-term together. Multiplying the results from Step 1, Step 2, and Step 3:
$\frac{3}{5} \times x^2 \times y = \frac{3x^2y}{5}$

Answer:

$\frac{3x^2y}{5}$