Problem 35 - College Algebra with Modeling and Visualization

Question

If possible, simplify the expression by hand. If you cannot, approximate the answer to the nearest hundredth. Variables represent any real number. $$ \sqrt[3]{(2 x)^{6}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 4x^2 \).

Step 1: Understand the Expression

The given expression is \( \sqrt[3]{(2x)^6} \). It represents the cube root of \((2x)^6\).

Step 2: Apply Exponent Rule

Recall the exponent rule: \((a^m)^n = a^{m \cdot n}\). Thus, \((2x)^6 = 2^6 \cdot x^6\).

Step 3: Combine Exponents with Root

By taking the cube root, \( \sqrt[3]{(2^6 \cdot x^6)} = \sqrt[3]{2^6} \cdot \sqrt[3]{x^6} \).

Step 4: Simplify Cube Roots

Simplify each cube root separately: \( \sqrt[3]{2^6} = 2^2 \) because \( 2^{6/3} = 2^2 \). Similarly, \( \sqrt[3]{x^6} = x^2 \) because \( x^{6/3} = x^2 \).

Step 5: Write Simplified Expression

Combine the simplified components: \( 2^2 \cdot x^2 = 4x^2 \). Therefore, the simplified expression is \( 4x^2 \).

Key Concepts

Cube Root

A cube root is the operation of finding a number which, when multiplied by itself three times, gives the original number. In mathematical terms, if you have a number or expression, and you want to find the cube root, you are looking for a value that, when cubed, returns the original number or expression.

Let's consider the concept of cube roots through the example given in the problem:

• The expression is \( \sqrt[3]{(2x)^6} \).
• The cube root operation is applied to the quantity \((2x)^6\).
• This involves understanding what the cube root signifies in terms of roots and powers.

For a simpler analogy, think of "cubing" a number, like 3, as \( 3 \times 3 \times 3 = 27 \). A cube root works in the reverse manner, asking "which number, when cubed, equals 27?" The answer is 3, thus \( \sqrt[3]{27} = 3 \).

In our specific case, the expression deals with a cube root of \( (2x)^6 \), which can be thought of as reversing a power of three.

Exponent Rules

Exponent rules help simplify expressions that involve exponents, such as powers of numbers or variables. Understanding these rules allows you to manage complex expressions with ease. Let's dive into some key aspects relevant to our exercise.When you have an expression in the form of \((a^m)^n\), exponent rules allow you to simplify it to \(a^{m \cdot n}\). This is a powerful tool when dealing with nested exponents.In our exercise:

• The expression \((2x)^6\) signifies both 2 and x raised to the 6th power.
• Apply exponent rules: \((2x)^6 = 2^6 \cdot x^6\).
• This breaks the expression into simpler parts that can be individually handled.

Another useful rule to apply is that of taking roots when exponents are involved. For example, the cube root of a square, such as \( (a^b)^c \), can be handled by adjusting the exponent using the root, leading to \( a^{b/c} \). This allows the expression to be simplified further, as seen when \( \sqrt[3]{x^6} \) becomes \( x^2 \).

Remembering these rules lets you deconstruct and reshape expressions efficiently.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations. They form the backbone of algebra and can appear simple or complex depending on the components and operations involved.In the given problem, the expression \((2x)^6\) is an algebraic expression that contains a base of \(2x\), which includes both a number and a variable. Here’s how you can look at such expressions:

• Identify the base and the exponent: \( (2x)\) is the base, and it's raised to the power of 6.
• Break it down: This expresses itself as a product \(2^6 \times x^6\) due to exponent rules.
• Coupling operations: Follow through with combining exponents or simplifying through operations like cube roots, as illustrated by our problem \( \sqrt[3]{2^6 \times x^6} \).

Simplifying algebraic expressions involves using rules of arithmetic and logic to reduce the expression to its most concise form. These expressions require careful handling of like terms and operations to achieve simplification, much as was done in solving the exercise to yield a simpler expression: \(4x^2\).

By understanding and manipulating algebraic expressions, you can tackle a wide range of mathematical problems.