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
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Set up the equation
To find the fourth root of 8, we can set up the equation:
$$ x^4 = 8 $$ -
Take the fourth root
To solve for $x$, we take the fourth root of both sides of the equation:
$$ x = \sqrt[4]{8} $$ -
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:
$$ x = \sqrt[4]{2^3} $$ -
Apply the exponent rule
Using the rule of exponents that states $ \sqrt[n]{a^m} = a^{\frac{m}{n}}$, we can further simplify:
$$ x = 2^{\frac{3}{4}} $$ -
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.
