Question

1. \\(\frac{24x^{5}y^{-7}}{8x^{-2}y^{2}}\\)

Explanation:

Step1: Simplify the coefficient

Divide the coefficient 24 by 8.
$\frac{24}{8} = 3$

Step2: Simplify the \(x\)-terms

Use the rule of exponents \( \frac{x^m}{x^n} = x^{m - n} \). For the \(x\)-terms, \(m = 5\) and \(n=-2\).
$x^{5 - (-2)} = x^{5 + 2} = x^7$

Step3: Simplify the \(y\)-terms

Use the rule of exponents \( \frac{y^m}{y^n} = y^{m - n} \). For the \(y\)-terms, \(m = -7\) and \(n = 2\).
$y^{-7 - 2} = y^{-9} = \frac{1}{y^9}$ (using the rule \(y^{-a}=\frac{1}{y^a}\))

Step4: Combine the results

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term together.
$3 \times x^7 \times \frac{1}{y^9} = \frac{3x^7}{y^9}$

Answer:

$\frac{3x^7}{y^9}$