91Ó°ÊÓ

The concept of the fourth root is a type of radical expression. When we see \(\sqrt[4]{x}\), we are looking for a number that, when multiplied by itself four times • gives us the original number \(x\).

To understand this better, take the number 16.
The fourth root of 16, written as \(\sqrt[4]{16}\), is 2 because \(2 \times 2 \times 2 \times 2 = 16\).

An important thing to remember is that the inside of the radical (the number we’re finding the root of) • must be positive when dealing with even roots, such as the fourth root. This is because any negative number raised to an even power • becomes positive.

Therefore, there is no real number that is the fourth root of a negative number, like -81. This makes the equation \(\sqrt[4]{-81} = -3\) impossible.

Exponents are a way to express repeated multiplication of the same number. For instance, \((-3)^4\) means multiplying -3 by itself four times, i.e., • \((-3) \times (-3) \times (-3) \times (-3)\).

Some helpful properties of exponents to remember:

• Multiplying a number by itself with an exponent is simply said as "raising the number."
• Raising a negative number to an even power results in a positive number.
• Exponents can also be used to express roots: for example, the fourth root can be written as an exponent \(x^{1/4}\).

When solving the equation \(\sqrt[4]{-81} = -3\), remember these properties:• Raising any real number to an even power cannot yield a negative result. That’s why any calculation like \((-3)^4\) will not give -81 but instead 81. Understanding these exponent properties helps to see why such a fourth root on a negative number doesn’t exist within the realm of real numbers.

Even powers, such as the second, fourth, sixth power, etc., • have a distinct characteristic: when a number (positive or negative) is raised to an even power, the result is always positive.

For instance, consider the number \(-2\).\((-2)^2\) is • \((-2) \times (-2) = 4\).Similarly, \((-3)^4\) is \((-3) \times (-3) \times (-3) \times (-3) = 81\).

• The negative signs cancel out with each paired multiplication, which makes the overall result positive. This cancellation property ensures that negative numbers become positive after being raised to even powers.

Due to this characteristic, it becomes evident why • the fourth root of a negative number cannot result in a negative number. Thus, calculations like \(\sqrt[4]{-81}\) equalling -3 are incorrect because they defy the rules of mathematics concerning even powers.