Question

follow the instructions for each problem. show all your work and express your answer in simplest form.

1. simplify:

(sqrt3{x^4y^2z^3})

1. multiply:

((sqrt{a} - sqrt{b})(sqrt{a} + sqrt{b}))

Explanation:

Problem 1: Simplify $\sqrt[3]{x^4 y^2 z^3}$

Step 1: Split the radical

We can split the cube root into the product of cube roots of each factor: $\sqrt[3]{x^4} \cdot \sqrt[3]{y^2} \cdot \sqrt[3]{z^3}$.

Step 2: Simplify each cube root

• For $\sqrt[3]{x^4}$, we can write $x^4$ as $x^3 \cdot x$, so $\sqrt[3]{x^3 \cdot x} = \sqrt[3]{x^3} \cdot \sqrt[3]{x} = x\sqrt[3]{x}$.
• $\sqrt[3]{y^2}$ cannot be simplified further, so it remains $\sqrt[3]{y^2}$.
• For $\sqrt[3]{z^3}$, the cube root of $z^3$ is $z$, since $z^3 = z \cdot z \cdot z$.

Step 3: Combine the simplified terms

Multiply the simplified cube roots together: $x\sqrt[3]{x} \cdot \sqrt[3]{y^2} \cdot z = xz\sqrt[3]{x y^2}$.

Step 1: Use the difference of squares formula

The product of a sum and a difference, $(m - n)(m + n)$, is given by the difference of squares formula: $m^2 - n^2$. Here, $m = \sqrt{a}$ and $n = \sqrt{b}$.

Step 2: Apply the formula

Substitute $m = \sqrt{a}$ and $n = \sqrt{b}$ into the formula: $(\sqrt{a})^2 - (\sqrt{b})^2$.

Step 3: Simplify the squares

• $(\sqrt{a})^2 = a$ (since the square of a square root of a number is the number itself, for non - negative $a$).
• $(\sqrt{b})^2 = b$ (for non - negative $b$).

Step 4: Subtract the simplified terms

$a - b$.

Answer:

$xz\sqrt[3]{x y^2}$

Problem 2: Multiply $(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})$