Simplify ((x^2y)^-4)/((xy)^-3)

Solution:

Step 1: Simplify the given expression.

Step 1.1

Utilize the rule for negative exponents, $b^{-n}=\frac{1}{b^n}$, to reposition ${\left(x^{2}y\right)}^{-4}$ to the denominator.

$$\frac{1}{{\left(xy\right)}^{-3}{\left(x^{2}y\right)}^4}$$

Step 1.2

Apply the negative exponent rule, $\frac{1}{b^{-n}}=b^n$, to move ${\left(xy\right)}^{-3}$ to the numerator.

$$\frac{{\left(xy\right)}^3}{{\left(x^{2}y\right)}^4}$$

Step 1.3

Invoke the product rule for exponents on $xy$.

$$\frac{x^3{y}^3}{{\left(x^{2}y\right)}^4}$$

Step 2: Simplify the denominator.

Step 2.1

Apply the product rule to $x^{2}y$.

$$\frac{x^3{y}^3}{{\left(x^2\right)}^4{y}^4}$$

Step 2.2

Handle the exponents in ${\left(x^2\right)}^4$.

Step 2.2.1

Use the power rule for exponents, ${\left(a^m\right)}^n=a^{mn}$.

$$\frac{x^3{y}^3}{x^{2\cdot 4}{y}^4}$$

Step 2.2.2

Perform the multiplication of $2$ by $4$.

$$\frac{x^3{y}^3}{x^8{y}^4}$$

Step 3: Reduce the expression by eliminating common factors.

Step 3.1

Eliminate the common $x^3$ factor from $x^3$ and $x^8$.

Step 3.1.1

Extract $x^3$ from $x^3{y}^3$.

$$\frac{x^3(y^3)}{x^8{y}^4}$$

Step 3.1.2

Remove the common factors.

Step 3.1.2.1

Factor out $x^3$ from $x^8{y}^4$.

$$\frac{x^3(y^3)}{x^3(x^5{y}^4)}$$

Step 3.1.2.2

Eliminate the shared factor.

$$\frac{\cancel{x^3}{y}^3}{\cancel{x^3}(x^5{y}^4)}$$

Step 3.1.2.3

Rewrite the simplified expression.

$$\frac{y^3}{x^5{y}^4}$$

Step 3.2

Remove the common $y^3$ factor from $y^3$ and $y^4$.

Step 3.2.1

Introduce a multiplicative identity, $1$.

$$\frac{y^3\cdot 1}{x^5{y}^4}$$

Step 3.2.2

Cancel out the common factors.

Step 3.2.2.1

Factor out $y^3$ from $x^5{y}^4$.

$$\frac{y^3\cdot 1}{y^3(x^{5}y)}$$

Step 3.2.2.2

Eliminate the shared factor.

$$\frac{\cancel{y^3}\cdot 1}{\cancel{y^3}(x^{5}y)}$$

Step 3.2.2.3

Present the final simplified expression.

$$\frac{1}{x^{5}y}$$

Knowledge Notes:

1. Negative Exponent Rule: The negative exponent rule states that for any nonzero number $b$ and any integer $n$, $b^{-n}=\frac{1}{b^n}$. This rule is used to transform expressions with negative exponents into their reciprocal forms with positive exponents.

2. Product Rule for Exponents: The product rule for exponents says that when multiplying two powers that have the same base, you can add the exponents: $a^m\cdot a^n=a^{m+n}$.

3. Power Rule for Exponents: The power rule for exponents indicates that when raising a power to another power, you multiply the exponents: ${\left(a^m\right)}^n=a^{mn}$.

4. Simplifying Expressions: Simplifying expressions often involves applying exponent rules, combining like terms, and reducing fractions by eliminating common factors in the numerator and denominator.

5. Common Factors: When simplifying fractions, if the numerator and denominator share a common factor, it can be cancelled out to simplify the expression further. This is based on the principle that any number divided by itself equals one.