Question

1. factor the greatest common factor of ( 3x^2yz - 4zyx^2 + 2xyz^2 ).
2. simplify (factor if necessary): ( \frac{a^3b - 2a^2b^2}{a^2b} )

simplify. write the answers with all exponents positive.

1. ( \frac{m^4p^5}{p^{-3}m^6} )
2. ( \frac{xxx^3y^5y^{-2}}{x^{-3}yy^{-6}} )

Explanation:

Problem 19

Step1: Identify common factors

Find shared terms: $x,y,z$; coefficients GCD is 1.

Step2: Factor out common terms

Extract $xyz$ from each term.
$3x^2yz - 4zyx^2 + 2xyz^2 = xyz(3x - 4x + 2z)$

Step3: Simplify inside parentheses

Combine like terms.
$xyz(3x - 4x + 2z) = xyz(-x + 2z)$

Step1: Factor numerator terms

Factor $a^2b$ from numerator.
$a^3b - 2a^2b^2 = a^2b(a - 2b)$

Step2: Cancel common factors

Divide numerator by $a^2b$.
$\frac{a^2b(a - 2b)}{a^2b} = a - 2b$

Step1: Use exponent quotient rule

Subtract exponents for like bases.
$\frac{m^4}{m^6} = m^{4-6}$, $\frac{p^5}{p^{-3}} = p^{5-(-3)}$

Step2: Calculate exponents

Simplify each exponent term.
$m^{-2}p^{8}$

Step3: Make exponents positive

Rewrite negative exponent as reciprocal.
$\frac{p^8}{m^2}$

Answer:

$xyz(2z - x)$

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Problem 20