Problem 34 - Algebra A Combined Function

Question

Find each cube root. $$ \sqrt[3]{-64 x^{6}} $$

Step-by-Step Solution

Answer

The cube root of \(-64x^6\) is \(-4x^2\).

Step 1: Recognize the Cube Root

We need to find the cube root of the expression \(-64x^6\). The cube root means finding a value that when raised to the power of 3 gives us \(-64x^6\). We express this mathematically as \(\sqrt[3]{-64x^6}\).

Step 2: Separate the Cube Root Into Parts

We can separate the cube root of \(-64x^6\) into two parts: the cube root of \(-64\) and the cube root of \(x^6\). Therefore, \(\sqrt[3]{-64x^6} = \sqrt[3]{-64} \times \sqrt[3]{x^6}\).

Step 3: Calculate the Cube Root of -64

The cube root of \(-64\) is \(-4\) because \((-4)^3 = -4 \times -4 \times -4 = -64\). So, \(\sqrt[3]{-64} = -4\).

Step 4: Calculate the Cube Root of x^6

To find \(\sqrt[3]{x^6}\), we recall that this is equivalent to \((x^6)^{1/3}\). By applying the power rule \((a^m)^n = a^{mn}\), we simplify it to \(x^{6\times(1/3)} = x^2\). So, \(\sqrt[3]{x^6} = x^2\).

Step 5: Combine the Results

Now, combine the results of the two parts: \(-4\) and \(x^2\). Together, the cube root of the original expression is \(-4x^2\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are important in mathematics and appear in various situations, including in debt calculations or below zero temperatures. When dealing with negative numbers in mathematical expressions, it is important to understand some basic rules:

• Multiplying two negative numbers results in a positive number. For example, \((-4) \times (-4) = 16\).
• Multiplying a negative number with a positive number results in a negative number, such as \((-4) \times 4 = -16\).
• When considering the cube root of a negative number, like \(\sqrt[3]{-64}\), the result is also negative. This is because raising a negative number to a power of three, as in \((-4)^3\), results in a negative product. \(-4 \times -4 \times -4 = -64\).

Understanding these properties helps in solving expressions involving negative numbers efficiently.

Exponents

Exponents are a way to express repeated multiplication of the same number. For example, \(x^6\) means that the number \(x\) is multiplied by itself six times: \(x \times x \times x \times x \times x \times x\). Exponents are widely used in algebra to simplify expressions and solve equations effectively. Here are some key points about exponents:

• The base is the number that gets multiplied, while the exponent indicates how many times the base multiplies itself. In \(x^6\), \(x\) is the base and \(6\) is the exponent.
• When you multiply terms with the same base, you add the exponents: \(x^a \times x^b = x^{a+b}\).
• When you divide terms with the same base, you subtract the exponents: \(x^a / x^b = x^{a-b}\).
• Raising a power to another power, you multiply the exponents: \((x^a)^b = x^{a \cdot b}\).

These rules make calculations involving powers more manageable, like finding cube roots.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operational symbols like addition, subtraction, multiplication, and division. These expressions can represent real-world situations or abstract mathematical problems.

• In the expression \(-64x^6\), \(-64\) is the constant coefficient, and \(x^6\) is the variable term raised to a power.
• Understanding parts of an expression helps in manipulating it for simplification or solving equations.

Algebraic expressions can be simplified by following specific rules, such as combining like terms or applying the distributive property. In this exercise, knowing how to separate and manage each part of \(-64x^6\) is crucial for correctly calculating the cube root.

Power Rule

The power rule is a fundamental concept used when working with exponents. It simplifies operations that involve powers, making expressions easier to handle. The power rule states that when you have an exponent raised to another exponent, you multiply the exponents. This can be represented as \((a^m)^n = a^{m \cdot n}\).

Here’s how you apply it:

• Given \((x^6)^{1/3}\), you multiply the exponents: \(6 \times \frac{1}{3} = 2\), which simplifies to \(x^2\).

By understanding the power rule, you can easily simplify expressions and calculate roots, such as cube roots, accurately. It is particularly useful when dealing with powers within roots and helps combine multiple steps into straightforward calculations.