Please check.
(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
Your answers are correct.
(4xy^2)(3x^-4y^5) = 12x^-3y^7
To simplify this expression, we can multiply the coefficients (4 * 3 = 12) and then multiply the variables (x * x = x^2) and (y^2 * y^5 = y^7). Finally, we combine the coefficients and variables to get 12x^-3y^7.
(2x)^5 (3x)^2 = 288x^7
To simplify this expression, we can apply the power of a product rule. We raise each factor inside the parentheses to the power outside the parentheses. So, (2x)^5 becomes (2^5)(x^5) = 32x^5, and (3x)^2 becomes (3^2)(x^2) = 9x^2. Then, we multiply the coefficients (32 * 9 = 288) and combine the variables to get 288x^7.
(-3x^2)^4 = 81x^8
To simplify this expression, we apply the power of a power rule. We raise the base (-3x^2) to the power outside the parentheses (4). So, (-3x^2)^4 becomes (-3)^4(x^2)^4 = 81x^8.
The set of integers is not a subset of the set of natural numbers because not all integers are included in the set of natural numbers. The set of natural numbers only includes positive whole numbers (1, 2, 3, ...), while the set of integers also includes negative whole numbers and zero.
All numbers are real numbers because the set of real numbers includes all rational and irrational numbers.
To check the first expression:
(4xy^2)(3x^-4y^5)
To simplify this, we can multiply the coefficients (4 * 3 = 12) and then apply the exponent rules for variables with the same base:
(x * x) = x^2
(y^2 * y^5) = y^(2 + 5) = y^7
Therefore, the simplified expression is:
12x^-3y^7
To check the second expression:
(2x)^5 (3x)^2
We can simplify this by applying the exponent rule for the power of a power:
(2x)^5 = 2^5 * (x)^5 = 32x^5
(3x)^2 = 3^2 * (x)^2 = 9x^2
Now, we can multiply the simplified expressions:
32x^5 * 9x^2 = 288x^7
Therefore, the simplified expression is:
288x^7
To check the third expression:
(-3x^2)^4
We can simplify this by applying the exponent rule for a negative base raised to an even power:
(-3x^2)^4 = 3^4 * (x^2)^4 = 81x^8
Therefore, the simplified expression is:
81x^8
As for the true or false statements:
The set of integers is not a subset of the set of natural numbers because the set of natural numbers includes positive integers starting from 1, while the set of integers includes negative numbers and zero.
All numbers are considered real numbers, so the statement is true.