(4xy^2)(3x^-4y^5) = 12x^-3y^7
(2x)^5 (3x)^2 = 288x^7
(-3x^2)^4 = 81x^8
True or False: the set of integers is a subset of the set of natural numbers.
False
All numbers are real numbers.
True
F, T, T, F (natural numbers are a subset of integers), F
To solve the first equation, (4xy^2)(3x^-4y^5) = 12x^-3y^7, we can multiply the coefficients and keep the bases (x and y) and add the exponents:
4 * 3 = 12
x * x^-4 = x^(-3 + (-4)) = x^-7
y^2 * y^5 = y^(2 + 5) = y^7
So, the final answer is 12x^-7y^7.
To solve the second equation, (2x)^5 (3x)^2, we can simplify the expressions inside the parentheses first:
(2x)^5 = 2^5 * x^5 = 32x^5
(3x)^2 = 3^2 * x^2 = 9x^2
Now, we can multiply the coefficients and keep the base (x) and add the exponents:
32 * 9 = 288
x^5 * x^2 = x^(5 + 2) = x^7
So, the final answer is 288x^7.
To solve the third equation, (-3x^2)^4, we need to raise the entire expression inside the parentheses to the 4th power:
(-3x^2)^4 = (-3)^4 * (x^2)^4 = 81x^8
So, the final answer is 81x^8.
To determine if the set of integers is a subset of the set of natural numbers, we need to understand their definitions. The set of natural numbers, denoted by N, includes all positive whole numbers starting from 1 (1, 2, 3, 4, ...). On the other hand, the set of integers, denoted by Z, includes both positive and negative whole numbers (-∞, ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ..., ∞).
Since the set of integers includes numbers like -1 and -2, which are not part of the set of natural numbers, the statement "the set of integers is a subset of the set of natural numbers" is False.
Moving on to the statement "All numbers are real numbers," it is indeed True. Real numbers encompass all rational and irrational numbers, including integers, fractions, decimals, and square roots, among others. Therefore, all numbers fall under the category of real numbers.