The cube root of 1000 requires finding the base that creates 1000 when multiplied by itself three times. 10 * 10 * 10 equals 1000, so the answer is 10.

Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. Therefore, the expression is rewritten as $6 \times \frac{3}{2}$. Treating the whole number 6 as $\frac{6}{1}$, we multiply the numerators ($6 \times 3 = 18$) and the denominators ($1 \times 2 = 2$). The resulting fraction is $\frac{18}{2}$, which simplifies to the whole number 9.

When multiplying two fractions, the numerators are multiplied by each other, and the denominators are multiplied by each other. For this problem, you multiply the numerators $2 \times 4$ to get 8. Then, you multiply the denominators $7 \times 5$ to get 35. The resulting product is $\frac{8}{35}$. This fraction is already in its simplest form as 8 and 35 share no common factors other than 1.

To multiply a fraction by a whole number, the whole number is treated as a numerator over 1 (i.e., 4 = $\frac{4}{1}$). You then multiply the numerators together ($3 \times 4 = 12$) and the denominators together ($5 \times 1 = 5$). This results in the fraction $\frac{12}{5}$. The other options represent common errors, such as multiplying the denominator by the whole number or adding instead of multiplying.

Because the entire expression (2xy) is enclosed in parentheses and raised to the power of zero, the zero exponent rule applies to the whole term, resulting in 1.

Applying the power rule, we multiply the exponent inside the parentheses by the exponent outside. The product of 5 and 2 is 10, so the expression simplifies to y^10.

Any non-zero variable or number raised to the zero power simplifies to exactly 1. Thus, as long as x is not zero, x^0 = 1.

To remove the radical from the denominator, multiply both numerator and denominator by sqrt(2). This results in (1 * sqrt(2)) / (sqrt(2) * sqrt(2)), which simplifies to sqrt(2) / 2.

To multiply expressions with identical bases, you must add the exponents while keeping the base unchanged. Adding 2 and 5 gives an exponent of 7, resulting in 5^7.

To evaluate the cube root of 512, we identify the number that gives 512 when cubed. Since 8 * 8 * 8 = 512, the answer is 8.