does ((five times x) times (the cube root of (five times x))) equal ((five to the 4/3 power) times (x to the 4/3 power))

Yes, the expression \(\left(5x \cdot \sqrt[3]{5x}\right)\) is equal to \(\left(5^{4/3} \cdot x^{4/3}\right)\).

Here's the step-by-step explanation:

1. **Start with the expression**:
   \[
   \left(5x \cdot \sqrt[3]{5x}\right)
   \]

2. **Rewrite the cube root** using fractional exponents:
   \[
   5x \cdot (5x)^{1/3}
   \]

3. **Apply the property of exponents**: \(a^m \cdot a^n = a^{m+n}\) to combine the terms:
   \[
   (5 \cdot 5^{1/3}) \cdot (x \cdot x^{1/3})
   \]
   This simplifies to:
   \[
   5^{1 + 1/3} \cdot x^{1 + 1/3}
   \]

4. **Simplify the exponents**:
   \[
   5^{4/3} \cdot x^{4/3}
   \]

Thus, \(\left(5x \cdot \sqrt[3]{5x}\right)\) simplifies to \(\left(5^{4/3} \cdot x^{4/3}\right)\), as you stated.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the general formula for multiplying numbers with cube roots?
2. How do you simplify expressions with fractional exponents?
3. What are the properties of exponents that are useful in algebraic manipulations?
4. How does one convert between radical notation and fractional exponents?
5. How would the expression change if the cube root were replaced by a square root?

**Tip:** When simplifying expressions with exponents and roots, always try to rewrite radicals as fractional exponents. This approach often makes it easier to apply exponent rules.

Explore whether the equation 5x * cube root of 5x equals 5 to the power of 4/3 times x to the power of 4/3 with a detailed step-by-step explanation. Learn about exponents, cube roots, and fractional exponents in algebra. Suitable for high school students.

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