Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$-\sqrt[3]{64}$$

Step-by-Step Solution

Answer

The value of the expression is -4.

Step 1: Identify the Expression

The given expression is \[-\sqrt[3]{64}\]You need to find the cube root of 64 and then apply the negative sign.

Step 2: Find the Cube Root of 64

The cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\). We need to find a number \(y\) such that \[y^3 = 64\]Finding basic cube roots, we find:\[4^3 = 4 \times 4 \times 4 = 64\]Thus, \(\sqrt[3]{64} = 4\).

Step 3: Apply the Negative Sign

Now that we have \(\sqrt[3]{64} = 4\), the original expression includes a negative sign:\[-\sqrt[3]{64} = -4\]Therefore, the value of the expression is -4.

Key Concepts

Negative Numbers

Negative numbers are a fundamental part of mathematics, representing values less than zero. These numbers are found to the left of zero on a number line. When dealing with negative numbers, it is important to remember these key points:

• The sign of a number, negative or positive, affects calculations. For instance, multiplying or dividing two negative numbers results in a positive number.
• Addition of a negative number is equivalent to subtracting the corresponding positive number. For example, \(-2 + 3\) is the same as \(3 - 2\).
• When applying operations such as cube roots to negative numbers, pay attention to the placement of the negative sign. As shown in this exercise, the negative sign is applied after finding the cube root of the positive number, not before the operation.

Understanding these basic properties of negative numbers can provide a strong foundation in solving complex problems.

Radicals

Radicals are expressions that involve roots, such as square roots or cube roots. The radical sign \(\sqrt{}\) is commonly used to denote these roots. Specifically, a cube root \(\sqrt[3]{}\) represents a factor that, when raised to the power of three, gives the original number. Here's what you need to know about radicals:

• The expression \(\sqrt[3]{64}\) is asking us to find a number which, when cubed (multiplied by itself three times), equals 64.
• The calculation involved is \(4^3 = 4 \times 4 \times 4 = 64\), thus, \(\sqrt[3]{64} = 4\).
• Radicals can be simplified based on factoring the number under the radical sign, making calculations easier.

Because radicals can represent both positive and negative roots when dealing with odd roots (like cube roots), it's important to complete calculations step-by-step for accuracy.

Exponents

Exponents are shorthand notation used to indicate repeated multiplication of a number by itself. In a term like \(y^3\), the number 3 is the exponent which tells us how many times to multiply the base \(y\) by itself. Key points to consider about exponents include:

• An expression like \(y^3\) represents \(y \times y \times y\). For instance, \(4^3 = 4 \times 4 \times 4 = 64\).
• Exponents play a crucial role in determining cube and other roots, as they provide a clear representation of the factorization of a number.
• Properties of exponents, like \((a^m)(a^n) = a^{m+n}\), allow simplification of expressions in algebraic processes.
• Negative exponents are used but represent reciprocals, such as \(a^{-n} = \frac{1}{a^n}\).

Understanding exponents aids in simplifying numbers and solving complex equations efficiently, as seen in adjusting the cube root expression in this exercise.