To multiply terms with radicals, multiply the coefficients and radicands separately:

6\( \sqrt{8} \) x 8\( \sqrt{4} \)
(6 x 8)\( \sqrt{8 \times 4} \)
48\( \sqrt{32} \)

Now we need to simplify the radical:

48\( \sqrt{32} \)
48\( \sqrt{2 \times 16} \)
48\( \sqrt{2 \times 4^2} \)
(48)(4)\( \sqrt{2} \)
192\( \sqrt{2} \)

To subtract these radicals together their radicands must be the same:

5\( \sqrt{20} \) - 6\( \sqrt{5} \)
5\( \sqrt{4 \times 5} \) - 6\( \sqrt{5} \)
5\( \sqrt{2^2 \times 5} \) - 6\( \sqrt{5} \)
(5)(2)\( \sqrt{5} \) - 6\( \sqrt{5} \)
10\( \sqrt{5} \) - 6\( \sqrt{5} \)

Now that the radicands are identical, you can subtract them:

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

To add these radicals together their radicands must be the same:

2\( \sqrt{20} \) + 2\( \sqrt{5} \)
2\( \sqrt{4 \times 5} \) + 2\( \sqrt{5} \)
2\( \sqrt{2^2 \times 5} \) + 2\( \sqrt{5} \)
(2)(2)\( \sqrt{5} \) + 2\( \sqrt{5} \)
4\( \sqrt{5} \) + 2\( \sqrt{5} \)

Now that the radicands are identical, you can add them together:

Use PEMDAS (Parentheses, Exponents, Multipy/Divide, Add/Subtract):

4 + (5 + 3) ÷ 4 x 5 - 3²
P: 4 + (8) ÷ 4 x 5 - 3²
E: 4 + 8 ÷ 4 x 5 - 9
MD: 4 + \( \frac{8}{4} \) x 5 - 9
MD: 4 + \( \frac{40}{4} \) - 9
AS: \( \frac{16}{4} \) + \( \frac{40}{4} \) - 9
AS: \( \frac{56}{4} \) - 9
AS: \( \frac{56 - 36}{4} \)
\( \frac{20}{4} \)
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