Question

question 14
assume all variables are positive, and find the following root.
\\(\sqrt5{243a^5b^{15}}\\)
question 15
simplify the expression. assume that all variables represent positive numbers.
\\(\sqrt3{10a^3b^2} \cdot \sqrt3{25a^2b}\\)
question 16
identify the following then give the degree and the leading coefficient.
\\(5y^4 - 2y^3 + 5y^2 + 2y\\)
select an answer with degree and leading coefficient

Explanation:

Question 14

Step1: Factor radicand into powers

$243 = 3^5$, so $\sqrt[5]{3^5a^5b^{15}}$

Step2: Simplify each radical term

$\sqrt[5]{3^5}=3$, $\sqrt[5]{a^5}=a$, $\sqrt[5]{b^{15}}=b^3$

Step3: Combine simplified terms

$3 \cdot a \cdot b^3$

Question 15

Step1: Combine radicals (same index)

$\sqrt[5]{10a^3b^2 \cdot 25a^2b}$

Step2: Multiply inside the radical

$\sqrt[5]{(10 \cdot 25)a^{3+2}b^{2+1}} = \sqrt[5]{250a^5b^3}$

Step3: Simplify the radical

$\sqrt[5]{a^5} \cdot \sqrt[5]{250b^3} = a\sqrt[5]{250b^3}$

Question 16

Step1: Identify polynomial type

The expression is a polynomial in $y$.

Step2: Find highest exponent

The term $5y^4$ has exponent 4.

Step3: Find leading coefficient

Coefficient of $5y^4$ is 5.

Answer:

1. $3ab^3$
2. $a\sqrt[5]{250b^3}$
3. Polynomial, degree $\boldsymbol{4}$, leading coefficient $\boldsymbol{5}$