$$ -3 \sqrt[3]{5 x} $$ and $$ 4 \sqrt{5 x} $$

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UpStudy AI · Accepted Answer

It looks like you've provided two mathematical expressions:

1. $$ -3 \sqrt[3]{5x} $$
2. $$ 4 \sqrt{5x} $$

I'm going to assume you might be interested in solving the equation where these two expressions are equal:

$$
-3 \sqrt[3]{5x} = 4 \sqrt{5x}
$$

Let's solve for $$ x $$:

### Step 1: Express in Exponent Form
First, rewrite the radicals using exponents:

$$
-3 (5x)^{1/3} = 4 (5x)^{1/2}
$$

### Step 2: Isolate the Variable
Divide both sides by $$ (5x)^{1/3} $$ to isolate the terms involving $$ x $$:

$$
-3 = 4 (5x)^{1/2 - 1/3} = 4 (5x)^{1/6}
$$

### Step 3: Solve for $$ (5x)^{1/6} $$
$$
(5x)^{1/6} = \frac{-3}{4}
$$

### Step 4: Analyze the Solution
The expression $$ (5x)^{1/6} $$ represents the sixth root of $$ 5x $$. In the realm of real numbers:

- The sixth root of a number is real only if the number itself is non-negative ($$ 5x \geq 0 $$).
- Additionally, the sixth root yields a non-negative result.

However, the right side of the equation is $$ \frac{-3}{4} $$, which is negative. This leads to a contradiction because a non-negative number cannot equal a negative number.

### Conclusion
There are **no real solutions** to the equation $$ -3 \sqrt[3]{5x} = 4 \sqrt{5x} $$.

---

**If you had a different intention with the expressions or need further assistance with another type of operation (like simplifying, graphing, etc.), please provide more details!**
There are no real solutions to the equation $$ -3 \sqrt[3]{5x} = 4 \sqrt{5x} $$.

\( -3 \sqrt[3]{5 x} \) and \( 4 \sqrt{5 x} \)

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