Question

Evaluate the expression without using a calculator. $$ -64^{1 / 3} $$

Short answer

-4

Step-by-step solution

Recognize Cube Root

Observe that \(-64^{1/3}\) means the cube root of -64. It's called the cube root because we are trying to find a number that, when cubed (multiplied by itself twice), gives us -64.

Calculating Cube Root

Try to think of numbers whose cube could be 64. Remember that since the initial number is negative, the cube root will also need to be negative. In this case, the number is -4, because when -4 is cubed (-4*-4*-4), it gives -64.

Write Final Answer

Having found that the cube root of -64 is -4, write that as your final answer.

Key concepts

Exponentiation

Understanding exponentiation is crucial when dealing with cube roots, since it's fundamentally the process of raising numbers to a certain power. Exponentiation comes in handy, for example, when you're looking to cube a number. The notation for cubing a number, say 4, is written as 4^3, which simply means 4 multiplied by itself three times: 4 * 4 * 4, resulting in 64. In the case of negative numbers, such as -4, cubing would involve multiplying -4 * -4 * -4, leading to -64.

The inverse operation of exponentiation is finding the root, and with cube roots, we are specifically looking for what number, when cubed (raised to the third power), gives us the original number. In our homework problem, -64 was the given number, and we applied the concept of exponentiation inversely to find the cube root.

Radical Expressions

A radical expression is an expression that includes a radical symbol, which in mathematics is represented by \( \sqrt{} \), with the index (which indicates the type of root) written just above the left-hand upper corner of the radical sign. When there's no number present, it's assumed to be 2, which means a square root. In the case of cubed roots, a small 3 is written in that top corner, like this: \( \sqrt[3]{} \).

Our exercise displayed the radical expression \( -64^{1 / 3} \), which is another way to express the cube root of -64. The fractional exponent 1/3 indicates that we are dealing with the third root, meaning that we're looking for the number that can be multiplied by itself twice to obtain -64. Handling radical expressions requires recognizing these symbols and understanding their meaning in the context of roots.

Simplifying Expressions

The process of simplifying expressions, particularly radical ones, involves reducing the expression to its most basic and comprehensible form. Simplifying may include getting rid of any radicals that can be evaluated or converting the expression into a form that's easier to compute without such symbols. Simplifying often requires the identification of perfect cubes within the radical, as we did with the cube root of -64.

In our problem, we found the cube root and simplified the radical expression \( -64^{1 / 3} \) to -4, without the need for any further steps or calculations. Remember that when simplifying cube roots, look for cubes that are factors of the radicand (the number under the root symbol). Once such a factor is isolated, use the cube root of that factor to simplify the expression. The goal is always to make the expression as clear and concise as possible, allowing for an easy computation and understanding.