Problem 76 - Technical Mathematics with Calculus

Question

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

Step-by-Step Solution

Answer

\( 250x\sqrt{2x} \)

Step 1: Apply the exponential rule

To raise a product to a power, raise each factor of the product to that power. For the expression \( (5\sqrt{2x})^3 \) we will raise both 5 and \(\sqrt{2x}\) to the power of 3 separately.

Step 2: Calculate the cube of 5

Calculate \(5^3\), which is \(5 \times 5 \times 5 = 125\).

Step 3: Calculate the cube of \(\sqrt{2x}\)

To cube \(\sqrt{2x}\), we calculate \( (\sqrt{2x})^3 = (\sqrt{2x}) \times (\sqrt{2x}) \times (\sqrt{2x}) = 2x \sqrt{2x} \) because \(\sqrt{2x}^2 = 2x\).

Step 4: Combine the results

Multiply the results of Step 2 and Step 3 to get the final answer. \( 125 \times (2x \sqrt{2x}) = 250x\sqrt{2x} \).

Key Concepts

Exponential Rules

Understanding exponential rules is crucial when manipulating numbers and variables in algebra. An exponent tells us how many times a number, known as the base, is multiplied by itself. For instance, in the expression \( a^n \), \( a \) is the base, and \( n \) is the exponent, or power. When dealing with products (a combination of numbers and/or variables multiplied together) raised to an exponent, like \( (5\sqrt{2x})^3 \), we apply one of the fundamental exponential rules: to raise a product to a power, raise each factor of the product to that power separately.

This rule is vital in simplifying expressions efficiently and correctly. For our example, we raise the number 5 and the radical expression \( \sqrt{2x} \) separately to the third power. That is, \( 5^3 \times (\sqrt{2x})^3 \). By understanding and correctly applying exponential rules, you can simplify expressions and solve algebra problems more effectively.

Cube Roots

The cube root of a number \( a \) is a number \( x \) such that \( x^3 = a \). It's the inverse operation of cubing a number. In algebra, recognizing when and how to apply the concept of cube roots can be essential in solving equations and simplifying radical expressions.

When we encounter an expression like \( (\sqrt{2x})^3 \) as in our example, we are effectively cubing a square root. In general, cubing a square root cancels one instance of the square root since \( (\sqrt{a})^2 = a \) and we are left with one instance of \( \sqrt{a} \) untouched. Hence, \( (\sqrt{a})^3 = a\sqrt{a} \). Utilizing this knowledge, the process of simplifying expressions that involve both cube and square root operations becomes much more straightforward.

Simplifying Square Roots

Simplifying square roots is a fundamental skill in algebra. A square root, \( \sqrt{a} \), asks what number multiplied by itself gives the number \( a \). However, when we deal with expressions under the radical sign, we look for ways to simplify the expression without altering its value. This often includes finding the square root of perfect squares, which are numbers that are squares of integers, and factoring expressions to pull out perfect squares.

In our challenge problem \( (5\sqrt{2x})^3 \), we encounter the square root in an exponentiated form. Simplifying such an expression requires us to first deal with the exponent, which results in a cube of the square root. When simplified, as per the steps provided, we find that \( (\sqrt{2x})^3 \) turns into \( 2x\sqrt{2x} \) due to the cubing operation detailed in earlier sections. Familiarity with simplifying square roots will not only help in scenarios like these but also when solving equations, understanding functions, and analyzing graphs.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. They're the language through which we express relationships in algebra. Simplifying algebraic expressions means to reduce them to their most basic form without changing their meaning or value. This involves combining like terms, applying distributive properties, factoring, and, as in our example, using the rules of exponents and radicals.

The original exercise \( (5\sqrt{2x})^3 \) is an algebraic expression that needs simplification. To do this, we use the processes discussed above, combining exponent rules with our understanding of square roots to simplify each part individually before finally combining them to find the value of the entire expression. The objective is always to end up with an expression that is easier to work with, whether it's for further algebraic manipulations or for graphing and interpreting in a real-world context. Keeping algebraic expressions as simple as possible helps to avoid errors and confusion further down the line in solving any algebraic problems.