Simplify ((z^2y^-1x^-3)/(x^-8z^6y^4))^-4

Solution:

Step:1

Apply the negative exponent rule $b^{-n} = \frac{1}{b^n}$ to rewrite $y^{-1}$ in the denominator: $\left(\frac{z^2 x^{-3}}{x^{-8} z^6 y^4 \cdot y}\right)^{-4}$.

Step:2

Combine $y^4$ and $y$ by adding their exponents.

Step:2.1

Relocate $y$: $\left(\frac{z^2 x^{-3}}{x^{-8} z^6 (y \cdot y^4)}\right)^{-4}$.

Step:2.2

Combine $y$ and $y^4$.

Step:2.2.1

Express $y$ as $y^1$: $\left(\frac{z^2 x^{-3}}{x^{-8} z^6 (y^1 \cdot y^4)}\right)^{-4}$.

Step:2.2.2

Apply the power rule $a^m a^n = a^{m+n}$ to sum the exponents: $\left(\frac{z^2 x^{-3}}{x^{-8} z^6 y^{1+4}}\right)^{-4}$.

Step:2.3

Add the exponents $1$ and $4$: $\left(\frac{z^2 x^{-3}}{x^{-8} z^6 y^5}\right)^{-4}$.

Step:3

Move $x^{-8}$ to the numerator using the negative exponent rule $\frac{1}{b^{-n}} = b^n$: $\left(\frac{z^2 x^{-3} \cdot x^8}{z^6 y^5}\right)^{-4}$.

Step:4

Combine $x^{-3}$ and $x^8$ by adding their exponents.

Step:4.1

Relocate $x^8$: $\left(\frac{z^2 (x^8 \cdot x^{-3})}{z^6 y^5}\right)^{-4}$.

Step:4.2

Use the power rule to sum the exponents: $\left(\frac{z^2 x^{8-3}}{z^6 y^5}\right)^{-4}$.

Step:4.3

Subtract $3$ from $8$: $\left(\frac{z^2 x^5}{z^6 y^5}\right)^{-4}$.

Step:5

Cancel out the common factor of $z^2$ and $z^6$.

Step:5.1

Factor out $z^2$ from $z^2 x^5$: $\left(\frac{z^2 (x^5)}{z^6 y^5}\right)^{-4}$.

Step:5.2

Cancel the common factors.

Step:5.2.1

Factor out $z^2$ from $z^6 y^5$: $\left(\frac{z^2 (x^5)}{z^2 (z^4 y^5)}\right)^{-4}$.

Step:5.2.2

Cancel the common factor: $\left(\frac{\cancel{z^2} x^5}{\cancel{z^2} (z^4 y^5)}\right)^{-4}$.

Step:5.2.3

Rewrite the simplified expression: $\left(\frac{x^5}{z^4 y^5}\right)^{-4}$.

Step:6

Invert the fraction by changing the sign of the exponent: $\left(\frac{z^4 y^5}{x^5}\right)^4$.

Step:7

Distribute the exponent using the power rule $(ab)^n = a^n b^n$.

Step:7.1

Apply the power rule to $\frac{z^4 y^5}{x^5}$: $\frac{(z^4 y^5)^4}{(x^5)^4}$.

Step:7.2

Apply the power rule to $z^4 y^5$: $\frac{(z^4)^4 (y^5)^4}{(x^5)^4}$.

Step:8

Simplify the numerator by multiplying the exponents.

Step:8.1

Multiply the exponents in $(z^4)^4$.

Step:8.1.1

Apply the power rule $(a^m)^n = a^{mn}$: $\frac{z^{4 \cdot 4} (y^5)^4}{(x^5)^4}$.

Step:8.1.2

Compute $4 \times 4$: $\frac{z^{16} (y^5)^4}{(x^5)^4}$.

Step:8.2

Multiply the exponents in $(y^5)^4$.

Step:8.2.1

Apply the power rule $(a^m)^n = a^{mn}$: $\frac{z^{16} y^{5 \cdot 4}}{(x^5)^4}$.

Step:8.2.2

Compute $5 \times 4$: $\frac{z^{16} y^{20}}{(x^5)^4}$.

Step:9

Multiply the exponents in $(x^5)^4$.

Step:9.1

Apply the power rule $(a^m)^n = a^{mn}$: $\frac{z^{16} y^{20}}{x^{5 \cdot 4}}$.

Step:9.2

Compute $5 \times 4$: $\frac{z^{16} y^{20}}{x^{20}}$.

Knowledge Notes:

To solve the given problem, several algebraic rules and properties are used:

1. Negative Exponent Rule: $b^{-n} = \frac{1}{b^n}$, which allows us to move factors between the numerator and denominator by changing the sign of the exponent.

2. Power Rule for Exponents: $a^m a^n = a^{m+n}$, which is used when multiplying like bases; we add the exponents.

3. Power of a Power Rule: $(a^m)^n = a^{mn}$, which is used when raising a power to another power; we multiply the exponents.

4. Simplification of Fractions: When the same factor appears in both the numerator and denominator, it can be canceled out.

5. Distributive Property of Exponents: $(ab)^n = a^n b^n$, which allows us to apply an exponent to a product by raising each factor to the exponent separately.

These rules are systematically applied to simplify the given expression step by step.