Simplify (x^(1/2)y^-2z^4)/(x^(3/2)y^3z^(3/2))

Solution:

Step 1:

Apply the negative exponent rule $b^{n} = \frac{1}{b^{-n}}$ to transfer $x^{\frac{1}{2}}$ to the denominator: $\frac{y^{-2} z^{4}}{x^{\frac{3}{2}} y^{3} z^{\frac{3}{2}} x^{-\frac{1}{2}}}$.

Step 2:

Transfer $y^{-2}$ to the denominator using the negative exponent rule $b^{-n} = \frac{1}{b^{n}}$: $\frac{z^{4}}{x^{\frac{3}{2}} y^{3} z^{\frac{3}{2}} x^{-\frac{1}{2}} y^{2}}$.

Step 3:

Begin simplifying the denominator.

Step 3.1:

Combine $x^{\frac{3}{2}}$ and $x^{-\frac{1}{2}}$ by adding their exponents.

Step 3.1.1:

Rearrange to place $x^{-\frac{1}{2}}$ next to $x^{\frac{3}{2}}$: $\frac{z^{4}}{x^{-\frac{1}{2}} x^{\frac{3}{2}} y^{3} z^{\frac{3}{2}} y^{2}}$.

Step 3.1.2:

Apply the power rule $a^{m} a^{n} = a^{m + n}$: $\frac{z^{4}}{x^{-\frac{1}{2} + \frac{3}{2}} y^{3} z^{\frac{3}{2}} y^{2}}$.

Step 3.1.3:

Combine the exponents over a common denominator: $\frac{z^{4}}{x^{\frac{-1 + 3}{2}} y^{3} z^{\frac{3}{2}} y^{2}}$.

Step 3.1.4:

Add the exponents: $\frac{z^{4}}{x^{\frac{2}{2}} y^{3} z^{\frac{3}{2}} y^{2}}$.

Step 3.1.5:

Simplify the exponent by dividing: $\frac{z^{4}}{x^{1} y^{3} z^{\frac{3}{2}} y^{2}}$.

Step 3.2:

Reduce $x^{1}$ to $x$.

Step 3.3:

Combine $y^{3}$ and $y^{2}$ by adding their exponents.

Step 3.3.1:

Group $y^{2}$ and $y^{3}$ together: $\frac{z^{4}}{x ( y^{2} y^{3} ) z^{\frac{3}{2}}}$.

Step 3.3.2:

Apply the power rule $a^{m} a^{n} = a^{m + n}$: $\frac{z^{4}}{x y^{2 + 3} z^{\frac{3}{2}}}$.

Step 3.3.3:

Add the exponents: $\frac{z^{4}}{x y^{5} z^{\frac{3}{2}}}$.

Step 4:

Move $z^{\frac{3}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{-n}$: $\frac{z^{4} z^{-\frac{3}{2}}}{x y^{5}}$.

Step 5:

Combine $z^{4}$ and $z^{-\frac{3}{2}}$ by adding their exponents.

Step 5.1:

Use the power rule $a^{m} a^{n} = a^{m + n}$: $\frac{z^{4 - \frac{3}{2}}}{x y^{5}}$.

Step 5.2:

Express $4$ as a fraction with a common denominator by multiplying by $\frac{2}{2}$: $\frac{z^{4 \cdot \frac{2}{2} - \frac{3}{2}}}{x y^{5}}$.

Step 5.3:

Multiply $4$ by $\frac{2}{2}$: $\frac{z^{\frac{4 \cdot 2}{2} - \frac{3}{2}}}{x y^{5}}$.

Step 5.4:

Combine the numerators over the common denominator: $\frac{z^{\frac{4 \cdot 2 - 3}{2}}}{x y^{5}}$.

Step 5.5:

Simplify the numerator.

Step 5.5.1:

Multiply $4$ by $2$: $\frac{z^{\frac{8 - 3}{2}}}{x y^{5}}$.

Step 5.5.2:

Subtract $3$ from $8$: $\frac{z^{\frac{5}{2}}}{x y^{5}}$.

Knowledge Notes:

The problem-solving process involves simplifying a complex algebraic fraction by applying exponent rules. The relevant knowledge points include:

1. Negative Exponent Rule: For any non-zero base $b$ and any integer $n$, $b^{-n} = \frac{1}{b^n}$. This rule is used to move factors between the numerator and denominator of a fraction.

2. Power Rule for Exponents: When multiplying like bases, add the exponents: $a^m \cdot a^n = a^{m+n}$.

3. Simplifying Fractions: Combining like terms and simplifying expressions within fractions to reach the simplest form.

4. Common Denominators: When combining fractions or terms with exponents, it's often necessary to express whole numbers as fractions with a common denominator to combine them correctly.

5. Simplifying Exponents: After applying the rules for exponents, the expression is often simplified by performing the arithmetic within the exponents to reach the final simplified form.

Understanding these rules and how to apply them in sequence is crucial for simplifying algebraic expressions involving exponents.