Question

simplify the following expressions
a. ( (3x^{4}y^{-2})(2x^{4}y) )
b. ( \frac{18y^{7}x^{-2}}{3(x^{-4}y)^{3}} )

Explanation:

Part a

Step1: Multiply coefficients and like variables

Multiply the coefficients \(3\) and \(2\), and for \(x\) and \(y\) use the rule \(a^m \cdot a^n = a^{m + n}\).
\((3x^{4}y^{-2})(2x^{4}y)=3\times2\times x^{4 + 4}\times y^{-2+1}\)

Step2: Simplify each part

Calculate \(3\times2 = 6\), \(x^{4 + 4}=x^{8}\), \(y^{-2 + 1}=y^{-1}=\frac{1}{y}\) (but we can also write it with positive exponents or keep as is, here we combine):
\(6x^{8}y^{-1}=\frac{6x^{8}}{y}\) or \(6x^{8}y^{-1}\) (both are simplified, but usually positive exponents for \(y\) is preferred as \(\frac{6x^{8}}{y}\))

Part b

Step1: Simplify the denominator's power

First, expand \((x^{-4}y)^{3}\) using \((ab)^n=a^n b^n\) and \((a^m)^n=a^{mn}\).
\((x^{-4}y)^{3}=x^{-4\times3}y^{3}=x^{-12}y^{3}\)
So the denominator becomes \(3\times x^{-12}y^{3}\)

Step2: Simplify the fraction's coefficients and variables

The coefficient: \(\frac{18}{3}=6\)
For \(x\): use \(a^m\div a^n=a^{m - n}\), so \(x^{-2}\div x^{-12}=x^{-2-(-12)}=x^{10}\)
For \(y\): \(y^{7}\div y^{3}=y^{7 - 3}=y^{4}\)
Combine them: \(6\times x^{10}\times y^{4}\)

Step3: Write the final simplified form

\(6x^{10}y^{4}\)

Answer:

s:
a. \(\boldsymbol{\frac{6x^{8}}{y}}\) (or \(6x^{8}y^{-1}\))
b. \(\boldsymbol{6x^{10}y^{4}}\)