Question

46 simplify using laws of exponents. \\(\frac{4xy^2}{10x^4y^2} =\\) \\(\boldsymbol{\text{a}}\\) \\(\frac{2}{5x^3}\\) \\(\boldsymbol{\text{b}}\\) \\(\frac{2x^3}{5}\\) \\(\boldsymbol{\text{c}}\\) \\(\frac{2x^3y}{5}\\) \\(\boldsymbol{\text{d}}\\) \\(\frac{2y}{5x^3}\\)

Explanation:

Step1: Simplify the coefficients

Simplify the fraction of the coefficients \( \frac{4}{10} \). We can divide both the numerator and the denominator by their greatest common divisor, which is 2. So, \( \frac{4\div2}{10\div2}=\frac{2}{5} \).

Step2: Simplify the \( x \)-terms

Using the law of exponents \( \frac{a^m}{a^n}=a^{m - n} \) for \( x \)-terms. Here, \( m = 1 \) (from \( x^1 \) in the numerator) and \( n = 4 \) (from \( x^4 \) in the denominator). So, \( \frac{x}{x^4}=x^{1-4}=x^{-3}=\frac{1}{x^3} \).

Step3: Simplify the \( y \)-terms

Using the law of exponents \( \frac{a^m}{a^n}=a^{m - n} \) for \( y \)-terms. Here, \( m = 2 \) and \( n = 2 \). So, \( \frac{y^2}{y^2}=y^{2 - 2}=y^{0}=1 \) (since any non - zero number to the power of 0 is 1).

Step4: Combine the simplified terms

Multiply the simplified coefficient, \( x \)-term, and \( y \)-term together. We have \( \frac{2}{5}\times\frac{1}{x^3}\times1=\frac{2}{5x^3} \).

Answer:

A. \( \frac{2}{5x^3} \)