Question

1. simplify the expression

$(5x^{3}y^{-4})^{2}$

Explanation:

Step1: Apply power of a product rule

Distribute the exponent 2 to each factor inside the parentheses:
$(5x^3y^{-4})^2 = 5^2 \cdot (x^3)^2 \cdot (y^{-4})^2$

Step2: Calculate each raised factor

Compute the powers using the power rule $(a^m)^n = a^{m \cdot n}$:
$5^2 = 25$, $(x^3)^2 = x^{3 \cdot 2} = x^6$, $(y^{-4})^2 = y^{-4 \cdot 2} = y^{-8}$

Step3: Rewrite negative exponent

Convert the negative exponent to a positive one using $a^{-n} = \frac{1}{a^n}$:
$y^{-8} = \frac{1}{y^8}$

Step4: Combine all terms

Multiply the simplified factors together:
$25 \cdot x^6 \cdot \frac{1}{y^8} = \frac{25x^6}{y^8}$

Answer:

$\frac{25x^6}{y^8}$