Problem 8 - Algebra for College Students

Question

Evaluate each numerical expression. \(-64^{\frac{1}{3}}\)

Step-by-Step Solution

Answer

The cube root of -64 is -4.

Step 1: Understand the Expression

The expression \(-64^{\frac{1}{3}}\) represents the cube root of \(-64\). Cube roots are numbers that, when multiplied by themselves three times, equal the original number.

Step 2: Recognize the Base

Identify the base of the expression, which is \(-64\). This negative number is what we need to find the cube root for.

Step 3: Recall Properties of Cube Roots

Since cube roots of negative numbers are negative, look for a negative number which when cubed equals \(-64\). This is because \((-n)^3 = -n^3\).

Step 4: Calculate the Cube Root

Determine which number, when multiplied by itself three times equals \(64\). We know \((-4)\) works since \((-4) \times (-4) \times (-4) = -64\).

Step 5: Conclusion of Calculation

Thus, \(-4\) is the cube root of \(-64\). So, \(-64^{\frac{1}{3}} = -4\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, commonly used to represent loss, decrease, or deficit in various contexts. Understanding negative numbers is crucial when dealing with operations such as addition, subtraction, and exponentiation. When dealing with operations involving negative numbers, remember:

• The product of two negative numbers is positive.
• The product of a positive number and a negative number remains negative.

Having a grasp of these rules helps when evaluating expressions that include negative numbers, like in our example with \(-64^{\frac{1}{3}}\), where the number is negative, and we are looking for its cube root.

Exponentiation

Exponentiation refers to the mathematical operation involving two numbers - the base and the exponent. This operation is used to represent repeated multiplication of a number by itself. For instance, in the expression \(-64^{\frac{1}{3}}\), \(-64\) is the base and\(\frac{1}{3}\)is the exponent, indicating that we need to find the cube root of \(-64\).It is essential to note:

• An exponent of \(\frac{1}{n}\) means extracting the n-th root of the base number.
• When dealing with cube roots (exponent of \(\frac{1}{3}\)), you are determining which number multiplied by itself three times returns the original number.

Understanding exponentiation is vital in simplifying expressions and correctly applying mathematical operations to get accurate results.

Number Properties

Number properties encompass a range of rules and principles that dictate how numbers behave under various operations. Understanding these properties is fundamental in evaluating mathematical expressions accurately.Some key number properties to keep in mind are:

• Associative Property: The grouping of addition or multiplication does not affect the sum or product.
• Commutative Property: The order of addition or multiplication does not change the sum or product.
• Distributive Property: This combines both addition and multiplication, showing that a number times a sum equals the sum of individually multiplied numbers.

Application of these properties can simplify complex problems and help you make logical deductions when solving for cube roots, such as how we deduced that \(-4\) is the cube root of\(-64\). Knowing and applying number properties aids in a systematic approach to solving math problems.