Question

1. \\(\frac{6yx^{-2}}{(4yx^{2})^{2}}\\)

Explanation:

Step1: Expand the denominator

First, apply the power of a product rule \((ab)^n = a^n b^n\) and power rule \((a^m)^n = a^{mn}\) to the denominator:
$$(4yx^2)^2 = 4^2 y^2 (x^2)^2 = 16y^2x^4$$

Step2: Rewrite the expression

Substitute the expanded denominator back into the original fraction:
$$\frac{6yx^{-2}}{16y^2x^4}$$

Step3: Simplify coefficients

Reduce the numerical fraction by dividing numerator and denominator by their greatest common divisor (2):
$$\frac{6\div2}{16\div2} = \frac{3}{8}$$

Step4: Simplify \(y\)-terms

Use the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\):
$$\frac{y^1}{y^2} = y^{1-2} = y^{-1} = \frac{1}{y}$$

Step5: Simplify \(x\)-terms

Use the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\):
$$\frac{x^{-2}}{x^4} = x^{-2-4} = x^{-6} = \frac{1}{x^6}$$

Step6: Combine all simplified parts

Multiply the simplified coefficients, \(y\)-terms, and \(x\)-terms together:
$$\frac{3}{8} \cdot \frac{1}{y} \cdot \frac{1}{x^6} = \frac{3}{8y x^6}$$

Answer:

$\frac{3}{8x^6 y}$