Question

find the product.
$\left(\sqrt{6} + 6\
ight)\left(\sqrt{5} + 7\
ight)$
options:
$\sqrt{30} + 13\sqrt{5} + 42$
$\sqrt{30} + 42$
$\sqrt{30} + 7\sqrt{6} + 6\sqrt{5} + 42$
$14\sqrt{30} + 42$

Explanation:

Step1: Apply distributive property

Multiply each term in the first binomial by each term in the second binomial:
\((\sqrt{6} + 6)(\sqrt{5} + 7) = \sqrt{6} \cdot \sqrt{5} + \sqrt{6} \cdot 7 + 6 \cdot \sqrt{5} + 6 \cdot 7\)

Step2: Simplify each term

• \(\sqrt{6} \cdot \sqrt{5} = \sqrt{30}\) (product of square roots: \(\sqrt{a}\sqrt{b}=\sqrt{ab}\))
• \(\sqrt{6} \cdot 7 = 7\sqrt{6}\)
• \(6 \cdot \sqrt{5} = 6\sqrt{5}\)
• \(6 \cdot 7 = 42\)

Step3: Combine terms

Combine the simplified terms:
\(\sqrt{30} + 7\sqrt{6} + 6\sqrt{5} + 42\)

Answer:

\(\boldsymbol{\sqrt{30} + 7\sqrt{6} + 6\sqrt{5} + 42}\) (corresponding to the option " \(\boldsymbol{\sqrt{30} + 7\sqrt{6} + 6\sqrt{5} + 42}\) " among the choices)