Question

simplify. write the answer using only positive exponents

1. \\(\frac{36a^3 - 84a}{24a}=\\)
2. \\(-6x^{-3}y^{12}z^{14}\cdot15x^8y^{18}z^{-14}\\)
3. \\(\frac{72x^3y + 40xy^3}{8x}\\)

Explanation:

Problem 1: $\boldsymbol{\frac{36a^3 - 84a}{24a}}$

Step 1: Factor numerator and denominator

Factor out common terms: numerator has $12a$, denominator has $24a$.
$\frac{12a(3a^2 - 7)}{24a}$

Step 2: Simplify fraction

Cancel $12a$ from numerator and denominator.
$\frac{3a^2 - 7}{2}$

Step 1: Multiply coefficients and like bases

Multiply coefficients: $-6 \cdot 15 = -90$. For variables, use $a^m \cdot a^n = a^{m + n}$.
$-90x^{-3 + 8}y^{12 + 18}z^{14 + (-14)}$

Step 2: Simplify exponents

Simplify each exponent: $x^{5}$, $y^{30}$, $z^{0}$ (and $z^0 = 1$).
$-90x^5y^{30} \cdot 1$

Step 1: Split the fraction

Split into two fractions: $\frac{72x^3y}{8x} + \frac{40xy^3}{8x}$

Step 2: Simplify each fraction

Simplify coefficients and exponents: $\frac{72}{8}x^{3 - 1}y + \frac{40}{8}y^3$. Which is $9x^2y + 5y^3$.

Answer:

$\frac{3a^2 - 7}{2}$

Problem 2: $\boldsymbol{-6x^{-3}y^{12}z^{14} \cdot 15x^8y^{18}z^{-14}}$