Simplify (4x^3y^6z^4)/(8x^6yz^3)

Solution:

Step 1: Simplify the coefficients

• Divide the coefficients 4 and 8 to simplify the fraction.

• $\frac{4x^{3}y^{6}z^{4}}{8x^{6}yz^{3}}$ becomes $\frac{4}{8} \cdot \frac{x^{3}y^{6}z^{4}}{x^{6}yz^{3}}$.

• Reduce $\frac{4}{8}$ to $\frac{1}{2}$.

Step 2: Reduce the powers of x

• Simplify $x^{3}$ and $x^{6}$ by subtracting the exponents.

• $\frac{x^{3}}{x^{6}}$ becomes $x^{3-6}$ or $x^{-3}$.

Step 3: Reduce the powers of y

• Simplify $y^{6}$ and $y$ by subtracting the exponents.

• $\frac{y^{6}}{y}$ becomes $y^{6-1}$ or $y^{5}$.

Step 4: Reduce the powers of z

• Simplify $z^{4}$ and $z^{3}$ by subtracting the exponents.

• $\frac{z^{4}}{z^{3}}$ becomes $z^{4-3}$ or $z^{1}$.

Step 5: Combine the results

• Combine the simplified terms to get the final result.

• The simplified expression is $\frac{1}{2} \cdot x^{-3} \cdot y^{5} \cdot z$.

• Since $x^{-3}$ is in the denominator, the final simplified expression is $\frac{y^{5}z}{2x^{3}}$.

Knowledge Notes:

When simplifying algebraic fractions, the following knowledge points are relevant:

1. Reducing Fractions: A fraction is reduced by dividing both the numerator and the denominator by their greatest common factor.

2. Laws of Exponents: When dividing like bases with exponents, subtract the exponent in the denominator from the exponent in the numerator. For example, $a^{m} / a^{n} = a^{m-n}$.

3. Negative Exponents: A negative exponent indicates that the base is on the wrong side of the fraction line, so you flip it to the other side. For example, $x^{-n} = 1/x^{n}$.

4. Combining Terms: After simplifying the individual terms, combine them to form the final simplified expression.

5. Variables with Exponents: When variables have exponents, they are treated the same way as numerical bases with exponents during simplification.

By applying these principles systematically, one can simplify algebraic expressions effectively.