Math Problem Statement

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

Solution

The question in the image asks for which of the following expressions are equivalent to 61/364/36^{1/3} \cdot 6^{4/3}. Let's go through the process of solving this step by step:

  1. Simplifying the given expression:

    61/364/36^{1/3} \cdot 6^{4/3}

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

    61/3+4/3=65/36^{1/3 + 4/3} = 6^{5/3}

    The goal now is to see which of the listed options are equivalent to 65/36^{5/3}.


Examining the options:

  1. Option 1: 65/36^{5/3}

    This is exactly equal to 65/36^{5/3}, so this option is correct.

  2. Option 2: (65)1/3(6^5)^{1/3}

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

    (65)1/3=651/3=65/3(6^5)^{1/3} = 6^{5 \cdot 1/3} = 6^{5/3}

    This is also equivalent to 65/36^{5/3}, so this option is correct.

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

    Simplifying (6)3\left( \sqrt{6} \right)^3, where 6=61/2\sqrt{6} = 6^{1/2}, we get:

    (61/2)3=63/2\left( 6^{1/2} \right)^3 = 6^{3/2}

    This is not equal to 65/36^{5/3}, so this option is incorrect.

  4. Option 4: 63+643\sqrt[3]{6} + \sqrt[3]{6^4}

    This represents two separate terms: 63=61/3\sqrt[3]{6} = 6^{1/3} and 643=64/3\sqrt[3]{6^4} = 6^{4/3}. Adding them does not combine exponents, so this is not equivalent to 65/36^{5/3}. Hence, this option is incorrect.

  5. Option 5: 6643\sqrt[3]{6 \cdot 6^4}

    Simplifying inside the cube root:

    664=61+4=656 \cdot 6^4 = 6^{1+4} = 6^5

    Now applying the cube root:

    653=65/3\sqrt[3]{6^5} = 6^{5/3}

    This is equivalent to 65/36^{5/3}, so this option is correct.


Correct answers:

  • Option 1: 65/36^{5/3}
  • Option 2: (65)1/3(6^5)^{1/3}
  • Option 5: 6643\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 aman=am+na^m \cdot a^n = a^{m+n} and (am)n=amn(a^m)^n = a^{m \cdot n} to simplify terms efficiently.

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Math Problem Analysis

Mathematical Concepts

Exponents
Radicals
Simplification

Formulas

a^m * a^n = a^(m+n)
(a^m)^n = a^(m*n)
Cube root of a = a^(1/3)

Theorems

Exponent Multiplication Rule
Exponent Power Rule

Suitable Grade Level

Grades 9-11