91Ó°ÊÓ

When working with cube roots, you're essentially trying to find a number that, when multiplied by itself three times, gives you the original value. In mathematical terms, if you have a number \(a\), the cube root of \(a\) is a number \(b\) such that \(b \times b \times b = a\).

For example, in the expression \((-1000)^{2/3}\), the cube root of \(-1000\) is \(-10\) because \(-10 \times -10 \times -10 = -1000\). Notice that the cube root of a negative number is still negative, as you need three identical factors to result in a negative product.

To find a cube root easily:

• Break down the original number into prime factors.
• Re-group the factors into sets of three.
• Extract one factor from each group, allowing you to simplify the cube root.

Exponentiation is a math operation that involves raising a number, called the base, to the power of an exponent to express how many times the base is multiplied by itself. It's written as \(b^n\), where \(b\) is the base and \(n\) is the exponent.

For example, in the expression \((-10)^2\), \(-10\) serves as the base raised to the power of 2. This means you multiply \(-10\) by itself:

• \(-10 \times -10 = 100\)

This result is positive because multiplying two negative numbers yields a positive product. Exponentiation is crucial in simplifying expressions with rational exponents, such as when solving \((-1000)^{2/3}\).

Understanding exponentiation helps in handling complex problems as it allows for the evaluation of non-integer xponent values, breaking them into understandable steps.

Working with negative bases in mathematics can be tricky, especially under exponentiation. The key is to understand how the sign behaves under different scenarios.

When you raise a negative base to an integer exponent:

• Even exponent: The result is positive, because a negative number times itself an even number of times (such as in \((-10)^2\)) results in a positive number.
• Odd exponent: The result remains negative, because multiplying a negative number an odd number of times retains the negative sign.

These rules help when dealing with rational exponents, like in \((-1000)^{2/3}\). First, the cube root makes the base manageable, and then squaring the negative result makes it positive. Understanding how exponent rules apply to negative bases simplifies problem-solving with expressions involving rational exponents.