How can I simplify the expression involving the square root of 3 multiplied by the fifth root of 3?

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To simplify the expression √3 × &#8730[5]{3}, we first need to understand how to handle both the square root and the fifth root of the same number.

The square root of 3 can be expressed as:

  • √3 = 3^{1/2}

Similarly, the fifth root of 3 can be expressed as:

  • &#8730[5]{3} = 3^{1/5}

Now, we can combine these two expressions:

  • √3 × &#8730[5]{3} = 3^{1/2} × 3^{1/5}

When you multiply numbers with the same base, you can simply add their exponents:

  • 3^{1/2 + 1/5}

Next, we need to find a common denominator to add the fractions:

  • The common denominator for 2 and 5 is 10. Therefore:
  • 1/2 = 5/10 and 1/5 = 2/10

Now we can add the fractions:

  • 1/2 + 1/5 = 5/10 + 2/10 = 7/10

Putting it all together, we have:

  • 3^{7/10}

This means the simplified expression for √3 × &#8730[5]{3} is:

  • 3^{7/10} or &#8730[10]{3^7}

In conclusion, the expression √3 × &#8730[5]{3} can be simplified to:

  • 3^{7/10}

This result not only shows how to manipulate roots but also illustrates the power of combining exponents.

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