What is the fourth root of 8?

Understand the Problem

The question is asking for the fourth root of 8, which means we need to find a number that, when raised to the power of 4, equals 8. This involves understanding the concept of roots in mathematics.

Answer

The fourth root of 8 is \( 2^{\frac{3}{4}} \).
Answer for screen readers

The fourth root of 8 is ( 2^{\frac{3}{4}} ).

Steps to Solve

  1. Set up the equation
    To find the fourth root of 8, we can set up the equation:
    $$ x^4 = 8 $$

  2. Take the fourth root
    To solve for $x$, we take the fourth root of both sides of the equation:
    $$ x = \sqrt[4]{8} $$

  3. Simplify the fourth root
    We can express the number 8 in terms of powers:
    $$ 8 = 2^3 $$
    So we need to rewrite the fourth root:
    $$ x = \sqrt[4]{2^3} $$

  4. Apply the exponent rule
    Using the rule of exponents that states $ \sqrt[n]{a^m} = a^{\frac{m}{n}}$, we can further simplify:
    $$ x = 2^{\frac{3}{4}} $$

  5. Final form
    Thus, the fourth root of 8 in its simplest form is:
    $$ x = 2^{\frac{3}{4}} $$

The fourth root of 8 is ( 2^{\frac{3}{4}} ).

More Information

The fourth root of a number represents a value that, when raised to the power of 4, returns the original number. In this case, ( 2^{\frac{3}{4}} ) is a way to indicate that the number is between 1 and 2. It's a precise representation of the fourth root.

Tips

  • Mistaking roots for exponents: Remember that taking a root is the inverse operation to raising a power. Be careful not to confuse the order of operations.
  • Using approximation prematurely: When calculating roots, it’s helpful to express the answer in the simplest exact form before attempting to approximate it numerically.

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