Problem 78 - Technical Mathematics with Calculus

Question

Challenge Problems. Perform the indicated operation and simplify. $$(5 \sqrt[3]{2 a x})^{3}$$

Step-by-Step Solution

Answer

The simplified form is \( 125(2ax) = 250ax \)

Step 1: Apply the Power to a Product Rule

When an entire product is raised to a power, you can apply the power to each factor in the product individually. This uses the rule \( (ab)^n = a^n b^n \). So we apply the cube \( (3rd) \) power to both 5 and \( \sqrt[3]{2ax} \) separately.

Step 2: Cube the Coefficient

Cube the numerical coefficient 5 to get \( 5^3 = 125 \).

Step 3: Cube the Cube Root

When you raise a cube root to the third power, they cancel each other out. So \( (\sqrt[3]{2ax})^3 = 2ax \).

Step 4: Combine the Results

Combine the results of cubing the coefficient and cube root to get the final simplified form.

Key Concepts

Power to a Product Rule

When dealing with algebraic expressions, we often encounter situations where we need to raise a product of numbers or expressions to a power. The Power to a Product Rule is a key algebraic property that simplifies this process.

According to this rule, when a product like \( ab \) is raised to a power \( n \), you can apply the power to each factor individually, as in \( a^n b^n \). This is particularly useful when you're dealing with a mix of numerical coefficients and radicals, such as cube roots.

• It streamlines the process of exponentiation.
• It makes it easier to see the results of raising complex expressions to powers.
• It helps in identifying terms that can cancel or simplify further.

For example, the expression \( (5 \sqrt[3]{2ax})^3 \) can initially seem daunting. However, by applying the Power to a Product Rule, we break down the problem into smaller, manageable parts: cubing the 5 and separately cubing the \( \sqrt[3]{2ax} \), leading to \( 5^3(\sqrt[3]{2ax})^3 \). This simplification is the first step to making the problem more approachable.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent tells us how many times to multiply the base by itself. For instance, \( 5^3 \) means we need to multiply 5 by itself 3 times: \( 5 \times 5 \times 5 = 125 \).

This operation is fundamental in simplifying expressions, especially when dealing with powers and roots. When an expression involves a base that is a product or a fraction, exponentiation allows us to apply the power separately to each part of the product or to the numerator and denominator of a fraction.

• Understanding exponentiation simplifies complex problems.
• It reveals the underlying structure of polynomial expressions.
• Knowing how to properly exponentiate can prevent errors in calculation and simplification.

Exponentiation is not just for numbers but also applies to variables and more complicated expressions, enabling a wide range of algebraic manipulations and solutions.

Cube Roots

A cube root, denoted as \( \sqrt[3]{x} \), is a value that, when multiplied by itself three times, gives the original number \( x \). In other words, if \( y = \sqrt[3]{x} \), then \( y^3 = x \). Cube roots are the inverse operation of cubing a number.

When simplifying expressions like \( (\sqrt[3]{2ax})^3 \), understanding the nature of cube roots becomes crucial. In this case, the cube root and the cubing operation cancel each other out, leaving you with the expression under the cube root. This cancellation is true for all numbers and variables and is a direct consequence of the definition of cube roots.
For educational purposes, it's essential to note that:

• The cube root of a number can be positive or negative since a negative number cubed is still negative.
• While square roots are more commonly discussed, cube roots are also essential for solving a variety of mathematical problems.
• Recognizing when to apply the cancellation between cubing and taking a cube root can simplify many types of expressions.

Hence, when faced with the task of simplifying \( (\sqrt[3]{2ax})^3 \), one can directly write \( 2ax \) without further calculations.