https://file.aiquickdraw.com/mathbot/file/1725575579201yhmi9xmx.jpg

The question in the image asks for which of the following expressions are equivalent to \( 6^{1/3} \cdot 6^{4/3} \). Let's go through the process of solving this step by step:

1. **Simplifying the given expression:**

   \[
   6^{1/3} \cdot 6^{4/3}
   \]

   Using the rule of exponents \( a^m \cdot a^n = a^{m+n} \), we can add the exponents:

   \[
   6^{1/3 + 4/3} = 6^{5/3}
   \]

   The goal now is to see which of the listed options are equivalent to \( 6^{5/3} \).

---

### Examining the options:

1. **Option 1: \( 6^{5/3} \)**

   This is exactly equal to \( 6^{5/3} \), so this option is **correct**.

2. **Option 2: \( (6^5)^{1/3} \)**

   Using the exponent rule \( (a^m)^n = a^{m \cdot n} \), we get:

   \[
   (6^5)^{1/3} = 6^{5 \cdot 1/3} = 6^{5/3}
   \]

   This is also equivalent to \( 6^{5/3} \), so this option is **correct**.

3. **Option 3: \( \left( \sqrt{6} \right)^3 \)**

   Simplifying \( \left( \sqrt{6} \right)^3 \), where \( \sqrt{6} = 6^{1/2} \), we get:

   \[
   \left( 6^{1/2} \right)^3 = 6^{3/2}
   \]

   This is not equal to \( 6^{5/3} \), so this option is **incorrect**.

4. **Option 4: \( \sqrt[3]{6} + \sqrt[3]{6^4} \)**

   This represents two separate terms: \( \sqrt[3]{6} = 6^{1/3} \) and \( \sqrt[3]{6^4} = 6^{4/3} \). Adding them does not combine exponents, so this is not equivalent to \( 6^{5/3} \). Hence, this option is **incorrect**.

5. **Option 5: \( \sqrt[3]{6 \cdot 6^4} \)**

   Simplifying inside the cube root:

   \[
   6 \cdot 6^4 = 6^{1+4} = 6^5
   \]

   Now applying the cube root:

   \[
   \sqrt[3]{6^5} = 6^{5/3}
   \]

   This is equivalent to \( 6^{5/3} \), so this option is **correct**.

---

### Correct answers:
- Option 1: \( 6^{5/3} \)
- Option 2: \( (6^5)^{1/3} \)
- Option 5: \( \sqrt[3]{6 \cdot 6^4} \)

Would you like further details on any part of the process?

Here are 5 related questions to consider:
1. How do exponent rules apply to combining terms with the same base?
2. What is the general rule for simplifying expressions inside radicals?
3. How can you recognize when two expressions are equivalent using exponent laws?
4. What is the importance of cube roots in simplifying radical expressions?
5. How do fractional exponents relate to radicals?

**Tip**: Always look for exponent laws like \( a^m \cdot a^n = a^{m+n} \) and \( (a^m)^n = a^{m \cdot n} \) to simplify terms efficiently.

Which of the following expressions are equivalent to 6^(1/3) * 6^(4/3)? Select all that apply.

Learn how to simplify expressions involving exponents, such as 6^(1/3) * 6^(4/3), using exponent multiplication and power rules. This problem is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation