Simplify (12x^3y^2z)/(16xy^3z^2)
The question asks you to perform algebraic simplification on a given rational expression where both the numerator and the denominator are monomials. You need to reduce the expression to its simplest form by canceling common factors in the numerator and the denominator and by applying the laws of exponents correctly. The expression involves variables with exponents, which you will need to divide according to the exponent rules of division for powers with the same base.
$\frac{12 x^{3} y^{2} z}{16 x y^{3} z^{2}}$
Answer
Solution:
Step 1: Simplify the Numerical Coefficients
Identify and remove the greatest common divisor of the coefficients 12 and 16.
Step 1.1: Extract the Common Factor from the Numerator
Divide the numerator by the greatest common divisor.
$\frac{4 \cdot (3 x^{3} y^{2} z)}{16 x y^{3} z^{2}}$
Step 1.2: Simplify the Common Factors
Step 1.2.1: Extract the Common Factor from the Denominator
Divide the denominator by the same greatest common divisor.
$\frac{4 \cdot (3 x^{3} y^{2} z)}{4 \cdot (4 x y^{3} z^{2})}$
Step 1.2.2: Eliminate the Common Factor
Remove the common factor from both the numerator and denominator.
$\frac{\cancel{4} \cdot (3 x^{3} y^{2} z)}{\cancel{4} \cdot (4 x y^{3} z^{2})}$
Step 1.2.3: Present the Simplified Expression
Rewrite the fraction without the common factor.
$\frac{3 x^{3} y^{2} z}{4 x y^{3} z^{2}}$
Step 2: Simplify the Variable x
Remove common powers of the variable x from the numerator and denominator.
Step 2.1: Factor x from the Numerator
Isolate the common variable x in the numerator.
$\frac{x \cdot (3 x^{2} y^{2} z)}{4 x y^{3} z^{2}}$
Step 2.2: Simplify the Common Variables
Step 2.2.1: Factor x from the Denominator
Isolate the common variable x in the denominator.
$\frac{x \cdot (3 x^{2} y^{2} z)}{x \cdot (4 y^{3} z^{2})}$
Step 2.2.2: Eliminate the Common Variable
Remove the common variable from both the numerator and denominator.
$\frac{\cancel{x} \cdot (3 x^{2} y^{2} z)}{\cancel{x} \cdot (4 y^{3} z^{2})}$
Step 2.2.3: Present the Simplified Expression
Rewrite the fraction without the common variable.
$\frac{3 x^{2} y^{2} z}{4 y^{3} z^{2}}$
Step 3: Simplify the Variable y
Remove common powers of the variable y from the numerator and denominator.
Step 3.1: Factor y^2 from the Numerator
Isolate the common variable y^2 in the numerator.
$\frac{y^{2} \cdot (3 x^{2} z)}{4 y^{3} z^{2}}$
Step 3.2: Simplify the Common Variables
Step 3.2.1: Factor y^2 from the Denominator
Isolate the common variable y^2 in the denominator.
$\frac{y^{2} \cdot (3 x^{2} z)}{y^{2} \cdot (4 y z^{2})}$
Step 3.2.2: Eliminate the Common Variable
Remove the common variable from both the numerator and denominator.
$\frac{\cancel{y^{2}} \cdot (3 x^{2} z)}{\cancel{y^{2}} \cdot (4 y z^{2})}$
Step 3.2.3: Present the Simplified Expression
Rewrite the fraction without the common variable.
$\frac{3 x^{2} z}{4 y z^{2}}$
Step 4: Simplify the Variable z
Remove common powers of the variable z from the numerator and denominator.
Step 4.1: Factor z from the Numerator
Isolate the common variable z in the numerator.
$\frac{z \cdot (3 x^{2})}{4 y z^{2}}$
Step 4.2: Simplify the Common Variables
Step 4.2.1: Factor z from the Denominator
Isolate the common variable z in the denominator.
$\frac{z \cdot (3 x^{2})}{z \cdot (4 y z)}$
Step 4.2.2: Eliminate the Common Variable
Remove the common variable from both the numerator and denominator.
$\frac{\cancel{z} \cdot (3 x^{2})}{\cancel{z} \cdot (4 y z)}$
Step 4.2.3: Present the Simplified Expression
Rewrite the fraction without the common variable.
$\frac{3 x^{2}}{4 y z}$
Knowledge Notes:
To simplify a fraction involving algebraic expressions, follow these steps:
Identify and cancel out any common numerical factors from the numerator and denominator. This is done by finding the greatest common divisor (GCD) of the coefficients and dividing both by this number.
For each variable, cancel out common powers. If a variable appears in both the numerator and the denominator, you can reduce the expression by dividing out the common power of that variable. This is equivalent to subtracting the exponents of the common variable in the numerator and denominator.
Rewrite the simplified expression after each step to keep track of the changes and ensure accuracy.
The final expression should have no common factors in the numerator and denominator, and the variables should be reduced to their lowest possible powers.
Remember that when simplifying expressions, the order of operations and the properties of exponents are crucial. The properties of exponents include the product rule ($a^m \cdot a^n = a^{m+n}$), the quotient rule ($a^m / a^n = a^{m-n}$ when $n < m$), and the power rule ($(a^m)^n = a^{mn}$). These rules help in simplifying algebraic expressions efficiently.
