Combine the following expressions. (Assume any variables under an even root are nonnegative.)

Ask by Riley Savage. in the United States

Nov 17,2024

Question

Combine the following expressions. (Assume any variables under an even root are nonnegative.)

Ask by Riley Savage. in the United States

Nov 17,2024

UpStudy AI · Accepted Answer

To combine the given expressions, we will simplify each term step by step.

1. **First term: $$ 4 a \sqrt[4]{2 a^{3} b^{9} c^{4}} $$**

We can rewrite the fourth root:
   $$
   \sqrt[4]{2 a^{3} b^{9} c^{4}} = \sqrt[4]{2} \cdot \sqrt[4]{a^{3}} \cdot \sqrt[4]{b^{9}} \cdot \sqrt[4]{c^{4}}.
   $$
   Now, we simplify each part:
   - $$ \sqrt[4]{a^{3}} = a^{3/4} $$
   - $$ \sqrt[4]{b^{9}} = b^{9/4} $$
   - $$ \sqrt[4]{c^{4}} = c^{1} = c $$

Thus, we have:
   $$
   \sqrt[4]{2 a^{3} b^{9} c^{4}} = \sqrt[4]{2} \cdot a^{3/4} \cdot b^{9/4} \cdot c.
   $$
   Therefore, the first term becomes:
   $$
   4 a \cdot \sqrt[4]{2} \cdot a^{3/4} \cdot b^{9/4} \cdot c = 4 \sqrt[4]{2} \cdot a^{1 + 3/4} \cdot b^{9/4} \cdot c = 4 \sqrt[4]{2} \cdot a^{7/4} \cdot b^{9/4} \cdot c.
   $$

2. **Second term: $$ 2 b^{2} \sqrt[4]{162 a^{7} b c^{4}} $$**

Similarly, we rewrite the fourth root:
   $$
   \sqrt[4]{162 a^{7} b c^{4}} = \sqrt[4]{162} \cdot \sqrt[4]{a^{7}} \cdot \sqrt[4]{b} \cdot \sqrt[4]{c^{4}}.
   $$
   Now, we simplify each part:
   - $$ \sqrt[4]{162} = \sqrt[4]{81 \cdot 2} = \sqrt[4]{81} \cdot \sqrt[4]{2} = 3 \sqrt[4]{2} $$
   - $$ \sqrt[4]{a^{7}} = a^{7/4} $$
   - $$ \sqrt[4]{b} = b^{1/4} $$
   - $$ \sqrt[4]{c^{4}} = c $$

Thus, we have:
   $$
   \sqrt[4]{162 a^{7} b c^{4}} = 3 \sqrt[4]{2} \cdot a^{7/4} \cdot b^{1/4} \cdot c.
   $$
   Therefore, the second term becomes:
   $$
   2 b^{2} \cdot 3 \sqrt[4]{2} \cdot a^{7/4} \cdot b^{1/4} \cdot c = 6 \sqrt[4]{2} \cdot a^{7/4} \cdot b^{2 + 1/4} \cdot c = 6 \sqrt[4]{2} \cdot a^{7/4} \cdot b^{9/4} \cdot c.
   $$

3. **Third term: $$ \sqrt[4]{162 a^{7} b^{9} c^{4}} $$**

We can rewrite this as:
   $$
   \sqrt[4]{162 a^{7} b^{9} c^{4}} = \sqrt[4]{162} \cdot \sqrt[4]{a^{7}} \cdot \sqrt[4]{b^{9}} \cdot \sqrt[4]{c^{4}}.
   $$
   We already know:
   - $$ \sqrt[4]{162} = 3 \sqrt[4]{2} $$
   - $$ \sqrt[4]{a^{7}} = a^{7/4} $$
   - $$ \sqrt[4]{b^{9}} = b^{9/4} $$
   - $$ \sqrt[4]{c^{4}} = c $$

Thus, we have:
   $$
   \sqrt[4]{162 a^{7} b^{9} c^{4}} = 3 \sqrt[4]{2} \cdot a^{7/4} \cdot b^{9/4} \cdot c.
   $$

4. **Combining all terms:**

Now we can combine all three terms:
   $$
   4 \sqrt[4]{2} a^{7/4} b^{9/4} c + 6 \sqrt[4]{2} a^{7/4} b^{9/4} c + 3 \sqrt[4]{2} a^{7/4} b^{9/4} c.
   $$
   Factoring out $$ \sqrt[4]{2} a^{7/4} b^{9/4} c $$:
   $$
   (4 + 6 + 3) \sqrt[4]{2} a^{7/4} b^{9/4} c = 13 \sqrt[4]{2} a^{7/4} b^{9/4} c.
   $$

Thus, the combined expression is:
$$
\boxed{13 \sqrt[4]{2} a^{7/4} b^{9/4} c}.
$$
$$ \boxed{13 \sqrt[4]{2} a^{7/4} b^{9/4} c}. $$

Combine the following expressions. (Assume any variables under an even root are nonnegative.)

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