Question

use the rules of exponents to simplify the expression. \\(\frac{(-2x^2y)^3}{9x^2y^2}\\) resources read it submit answer 8. - / 1 points use the rules of exponents to evaluate the expression. \\((-2)^0\\) resources read it submit answer 9. - / 1 points use the rules of exponents to evaluate the expression. \\(5^{-2}\\)

Explanation:

Problem 1: Simplify $\boldsymbol{\frac{(-2x^2y)^3}{9x^2y^2}}$

Step1: Apply Power of a Product Rule

The power of a product rule states that $(ab)^n = a^n b^n$. So, we apply this to the numerator $(-2x^2y)^3$:
$$(-2x^2y)^3 = (-2)^3 \cdot (x^2)^3 \cdot y^3$$
Calculating each part: $(-2)^3 = -8$, $(x^2)^3 = x^{2 \cdot 3} = x^6$ (using the power of a power rule $(a^m)^n = a^{mn}$), and $y^3$ remains as is. So the numerator becomes $-8x^6y^3$.

Step2: Divide the Monomials

Now we have the expression $\frac{-8x^6y^3}{9x^2y^2}$. When dividing monomials with the same base, we use the quotient rule $a^m \div a^n = a^{m - n}$.

For the coefficients: $\frac{-8}{9}$ (they don't have a common factor, so we leave them as is).

For the $x$ terms: $x^6 \div x^2 = x^{6 - 2} = x^4$.

For the $y$ terms: $y^3 \div y^2 = y^{3 - 2} = y^1 = y$.

Putting it all together, we get $\frac{-8x^4y}{9}$.

Step1: Apply Zero Exponent Rule

The zero exponent rule states that any non - zero number raised to the power of 0 is 1. Since $-2
eq0$, we have:
$$(-2)^0 = 1$$

Step1: Apply Negative Exponent Rule

The negative exponent rule states that $a^{-n}=\frac{1}{a^n}$ (where $a
eq0$). So for $5^{-2}$, we can rewrite it as:
$$5^{-2}=\frac{1}{5^2}$$

Step2: Calculate the Denominator

We know that $5^2 = 5\times5 = 25$. So, $\frac{1}{5^2}=\frac{1}{25}$.

Answer:

$\boldsymbol{\frac{-8x^4y}{9}}$

Problem 2: Evaluate $\boldsymbol{(-2)^0}$