3. Simplify by combining like radicals. Write your answers in radical form (not in decimal form). $$ \begin{array}{l}14 \sqrt{5}+12 \sqrt{5} \ ext { b. } 3 \sqrt{405}+42 \sqrt{125}-11 \sqrt{20} \ ext { c. } \sqrt[3]{270}-32 \sqrt[8]{40}-11 \sqrt{49}\end{array} $$ d. $$ \sqrt[3]{96}-\sqrt[3]{75}+4 \sqrt[3]{64} $$

Question

3. Simplify by combining like radicals. Write your answers in radical form (not in decimal form). $$ \begin{array}{l}14 \sqrt{5}+12 \sqrt{5} \ 	ext { b. } 3 \sqrt{405}+42 \sqrt{125}-11 \sqrt{20} \ 	ext { c. } \sqrt[3]{270}-32 \sqrt[8]{40}-11 \sqrt{49}\end{array} $$ d. $$ \sqrt[3]{96}-\sqrt[3]{75}+4 \sqrt[3]{64} $$

UpStudy AI · Accepted Answer

Let's simplify each expression step by step.

### a. $$ 14 \sqrt{5} + 12 \sqrt{5} $$

Combine the like radicals:

$$
14 \sqrt{5} + 12 \sqrt{5} = (14 + 12) \sqrt{5} = 26 \sqrt{5}
$$

### b. $$ 3 \sqrt{405} + 42 \sqrt{125} - 11 \sqrt{20} $$

First, simplify each radical:

1. $$ \sqrt{405} = \sqrt{81 \cdot 5} = \sqrt{81} \cdot \sqrt{5} = 9 \sqrt{5} $$
2. $$ \sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5 \sqrt{5} $$
3. $$ \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5} $$

Now substitute back into the expression:

$$
3 \sqrt{405} = 3(9 \sqrt{5}) = 27 \sqrt{5}
$$
$$
42 \sqrt{125} = 42(5 \sqrt{5}) = 210 \sqrt{5}
$$
$$
-11 \sqrt{20} = -11(2 \sqrt{5}) = -22 \sqrt{5}
$$

Now combine:

$$
27 \sqrt{5} + 210 \sqrt{5} - 22 \sqrt{5} = (27 + 210 - 22) \sqrt{5} = 215 \sqrt{5}
$$

### c. $$ \sqrt[3]{270} - 32 \sqrt[8]{40} - 11 \sqrt{49} $$

First, simplify each term:

1. $$ \sqrt[3]{270} = \sqrt[3]{27 \cdot 10} = \sqrt[3]{27} \cdot \sqrt[3]{10} = 3 \sqrt[3]{10} $$
2. $$ \sqrt[8]{40} = \sqrt[8]{8 \cdot 5} = \sqrt[8]{8} \cdot \sqrt[8]{5} = 2^{3/8} \cdot 5^{1/8} $$ (this term is left in its radical form)
3. $$ \sqrt{49} = 7 $$

Now substitute back into the expression:

$$
\sqrt[3]{270} - 32 \sqrt[8]{40} - 11 \sqrt{49} = 3 \sqrt[3]{10} - 32 \sqrt[8]{40} - 77
$$

This expression cannot be simplified further in terms of combining like radicals.

### d. $$ \sqrt[3]{96} - \sqrt[3]{75} + 4 \sqrt[3]{64} $$

First, simplify each term:

1. $$ \sqrt[3]{96} = \sqrt[3]{32 \cdot 3} = \sqrt[3]{32} \cdot \sqrt[3]{3} = 2 \sqrt[3]{3} $$
2. $$ \sqrt[3]{75} = \sqrt[3]{25 \cdot 3} = \sqrt[3]{25} \cdot \sqrt[3]{3} = 5^{2/3} \cdot 3^{1/3} $$ (left in radical form)
3. $$ \sqrt[3]{64} = 4 $$

Now substitute back into the expression:

$$
\sqrt[3]{96} - \sqrt[3]{75} + 4 \sqrt[3]{64} = 2 \sqrt[3]{3} - \sqrt[3]{75} + 4(4) = 2 \sqrt[3]{3} - \sqrt[3]{75} + 16
$$

This expression cannot be simplified further in terms of combining like radicals.

### Final Answers:
- a. $$ 26 \sqrt{5} $$
- b. $$ 215 \sqrt{5} $$
- c. $$ 3 \sqrt[3]{10} - 32 \sqrt[8]{40} - 77 $$
- d. $$ 2 \sqrt[3]{3} - \sqrt[3]{75} + 16 $$
a. $$ 26 \sqrt{5} $$ b. $$ 215 \sqrt{5} $$ c. $$ 3 \sqrt[3]{10} - 32 \sqrt[8]{40} - 77 $$ d. $$ 2 \sqrt[3]{3} - \sqrt[3]{75} + 16 $$

Simplify by combining like radicals. Write your answers in radical form (not in decimal form).

d.

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