Question

find the least common multiple (lcm) of the following:

1. 45, 75, and 125
2. (5x^{5}y^{3}z) and (6x^{3}y^{4}z^{2})

add:

1. (\frac{7}{x^{2}+y}-\frac{4}{x^{2}+y}+\frac{3}{x^{2}+y})
2. (\frac{a}{4d^{2}}+\frac{5}{d}+\frac{b}{d^{3}})
3. factor the greatest common factor of (3x^{2}yz - 4zyx^{2}+2xyz^{2}).

Explanation:

Problem 15:

Step1: Prime factorize each number

$45=3^2\times5$, $75=3\times5^2$, $125=5^3$

Step2: Take max exponents of primes

LCM uses highest power of each prime: $3^2, 5^3$

Step3: Calculate the product

$\text{LCM}=3^2\times5^3=9\times125$

Step1: Identify common variables

For $x$: max exponent $5$; $y$: max exponent $3$; $z$: max exponent $2$; $d$: max exponent $3$

Step2: Multiply common terms

$\text{LCM}=5\times6\times x^5\times y^3\times z^2\times d^3$

Step3: Simplify the coefficient

$5\times6=30$

Step1: Combine like fractions

All terms have denominator $x^2+y$, so add numerators: $7-4+3$

Step2: Calculate numerator sum

$7-4+3=6$

Answer:

$1125$

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Problem 16: