Question

1. ( 7x^{8}y^{-2}z^{4} cdot 5x^{-3}y^{0}z^{3} )

Explanation:

Step1: Multiply the coefficients

Multiply the numerical coefficients \(7\) and \(5\).
\(7\times5 = 35\)

Step2: Combine \(x\) terms

Use the rule of exponents \(a^m\cdot a^n=a^{m + n}\) for \(x\) terms \(x^{8}\cdot x^{-3}\).
\(x^{8+( - 3)}=x^{5}\)

Step3: Combine \(y\) terms

Use the rule of exponents for \(y\) terms \(y^{-2}\cdot y^{0}\). Remember that \(a^{0}=1\) and \(a^m\cdot a^n=a^{m + n}\), so \(y^{-2+0}=y^{-2}=\frac{1}{y^{2}}\) (but we can keep it in exponent form for now)

Step4: Combine \(z\) terms

Use the rule of exponents for \(z\) terms \(z^{4}\cdot z^{3}\).
\(z^{4 + 3}=z^{7}\)

Step5: Combine all parts

Multiply the coefficient, \(x\), \(y\), and \(z\) terms together.
\(35\times x^{5}\times y^{-2}\times z^{7}=\frac{35x^{5}z^{7}}{y^{2}}\)

Answer:

\(\frac{35x^{5}z^{7}}{y^{2}}\)