Question

1. $sqrt3{(u^3 + 5)^{18}}$
2. $sqrt{196c^6d^4}$ 38. $sqrt{-64y^8z^6}$
3. $sqrt3{-27a^{15}b^9}$ 40. $sqrt4{-16x^{16}y^8}$

Explanation:

Let's solve each problem one by one:

Problem 35: $\sqrt[3]{(u^3 + 5)^{18}}$

Step 1: Recall the property of radicals $\sqrt[n]{a^m}=a^{\frac{m}{n}}$

For a cube root, $n = 3$, and the exponent of the base is $18$. So we can rewrite the radical as an exponent:
$$(u^3 + 5)^{\frac{18}{3}}$$

Step 2: Simplify the exponent

Simplify $\frac{18}{3}=6$. So the expression becomes:
$$(u^3 + 5)^6$$

Step 1: Recall the property of square roots $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ and $\sqrt{a^2}=a$ (for $a\geq0$)

First, find the square root of the constant term: $\sqrt{196} = 14$ (since $14^2=196$).
Then, for the variable terms:
For $c^6$, using the property $\sqrt{c^6}=c^{\frac{6}{2}} = c^3$ (since $\sqrt{x^m}=x^{\frac{m}{2}}$ for square root).
For $d^4$, $\sqrt{d^4}=d^{\frac{4}{2}}=d^2$.

Step 2: Combine the results

Multiply the square roots of the constant and variable terms together:
$$14\times c^3\times d^2 = 14c^3d^2$$

Step 1: Recall the property of imaginary numbers and square roots

We know that $\sqrt{-1}=i$. First, factor out the negative sign:
$\sqrt{-64y^8z^6}=\sqrt{-1\times64y^8z^6}$

Step 2: Use the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$

$\sqrt{-1}\times\sqrt{64y^8z^6}$
We know that $\sqrt{-1}=i$, $\sqrt{64} = 8$, $\sqrt{y^8}=y^{\frac{8}{2}}=y^4$, $\sqrt{z^6}=z^{\frac{6}{2}}=z^3$

Step 3: Combine the results

Multiply all the terms together:
$i\times8\times y^4\times z^3 = 8iy^4z^3$

Answer:

$(u^3 + 5)^6$

Problem 37: $\sqrt{196c^6d^4}$