Determine whether each of the following expressions is equivalent to x8/3
3√x^4 x 3√x^4 yes or no
(3√x)^0/(3√x^2)^-2 yes or no
((3√x)^4)^2 yes or no
8√x^3 yes or no
1. No. The expression 3√x^4 * 3√x^4 simplifies to (x^4)^(1/3) * (x^4)^(1/3). Using the property (a^m)^n = a^(m*n), this becomes x^(4/3) * x^(4/3) = x^(8/3), not x^(8/3).
2. Yes. The expression (3√x)^0 / (3√x^2)^-2 simplifies to 1/((x^(2/3))^2). Using the property (a/b)^c = a^c / b^c, this becomes 1 / (x^(4/3)), which is equivalent to x^(-4/3). Since x^(-4/3) can be rewritten as 1 / x^(4/3), and x^(8/3) is also equivalent to 1 / x^(4/3), these two expressions are equivalent.
3. Yes. The expression ((3√x)^4)^2 simplifies to (x^(4/3))^2. Using the property (a^b)^c = a^(b*c), this becomes x^(8/3), which is equivalent to x^(8/3).
4. No. The expression 8√x^3 simplifies to (x^3)^(1/8). Using the property (a^m)^n = a^(m*n), this becomes x^(3/8), which is not equivalent to x^(8/3).
A student states that −4x^5−3x^4+8x^7−6x
has four terms, a degree of 4, and a negative leading coefficient.
Is the student correct? Explain
No, the student is not correct. The given expression has five terms: -4x^5, -3x^4, 8x^7, -6x. The degree is determined by the term with the highest exponent of x, so in this case, the degree would be 7. The leading coefficient is the coefficient of the term with the highest exponent of x, so in this case, the leading coefficient is 8, not negative.
Victoria is finding the value of n for the equation 3√141/6=141/n
Victoria writes 3√141/6=(141/6)^3=14^3/6=141/2 so n =2
Is Victoria correct? Justify your answer.
No, Victoria is incorrect.
To solve the equation 3√(141/6) = 141/n, we need to apply the properties of radicals.
First, we can simplify the expression on the left side of the equation by rewriting 141/6 as (141)^(1/6).
Thus, we have 3√(141/6) = (141)^(1/6).
To solve for n, we need to isolate it on one side of the equation. We can do this by cubing both sides of the equation:
(3√(141/6))^3 = ((141)^(1/6))^3.
Simplifying further, we get:
141/6 = 141^(1/2) = √141.
Now we can solve for n by cross multiplying:
6 * √141 = 141 * n.
Dividing both sides by 141, we find:
n = (6 * √141) / 141.
Therefore, n is equal to (6 * √141) / 141, not 2.