Problem 112 - Elementary and Intermediate Algebra

Question

Look Alikes \(\cdots\) a. \(x \sqrt[3]{64 x^{6}}-x \sqrt[3]{x^{6}}\) b. \(x \sqrt{64 x^{6}}-x \sqrt{x^{6}}\)

Step-by-Step Solution

Answer

a. Simplifies to \(3x^3\); b. Simplifies to \(7x^4\).

Step 1: Simplifying Cube Roots

Let's simplify each term of the expression in part (a). The cube root of 64 is equal to 4, since \( 4^3 = 64 \). So, \( \sqrt[3]{64} = 4 \). For \( \sqrt[3]{x^6} \), the cube root of \( x^6 \) is \( x^2 \) because \( (x^2)^3 = x^6 \). Substitute these into the expression: \( x \cdot 4x^2 - x \cdot x^2 = 4x^3 - x^3 \).

Step 2: Factoring the Terms

Factor out the common factor \( x^3 \) from the simplified expression \( 4x^3 - x^3 \) to get \( (4 - 1)x^3 \).

Step 3: Simplify the Expression

Simplify the expression \( (4 - 1)x^3 = 3x^3 \).

Step 4: Simplifying Square Roots

For part (b), simplify each term separately. The square root of 64 is 8 since \( 8^2 = 64 \), so \( \sqrt{64} = 8 \). For \( \sqrt{x^6} \), the square root is \( x^3 \) because \( (x^3)^2 = x^6 \). Substitute back into the expression: \( x \cdot 8x^3 - x \cdot x^3 = 8x^4 - x^4 \).

Step 5: Factoring the Result

Factor out the common term \( x^4 \) from \( 8x^4 - x^4 \) to get \( (8 - 1)x^4 \).

Step 6: Simplify the Result

Simplify the expression \( (8 - 1)x^4 = 7x^4 \).

Key Concepts

Cube Roots

Understanding cube roots is essential when working with algebraic expressions that involve powers of three. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 64 is 4, because

• \(4 \times 4 \times 4 = 64\).

In the exercise, simplifying the cube root of an expression like \(x^6\) involves breaking down the power of the variable. Here, \( \sqrt[3]{x^6} \) becomes \(x^2\) because:

• \( (x^2)^3 = x^6\).

This helps in reducing the expression into simpler terms, making it easier to factor and solve.

Square Roots

Square roots might seem simpler than cube roots, but they still require careful handling to fully simplify expressions. A square root of a number is a value that, when multiplied by itself, equals the original number. Taking the example from the exercise, the square root of 64 is 8, since:

• \(8 \times 8 = 64\).

Similarly, for variables, the square root of \(x^6\) simplifies to \(x^3\). This is because:

• \((x^3)^2 = x^6\).

When you substitute these values back into the expression, it becomes easier to perform further operations such as factoring.

Factoring Expressions

Factoring is a crucial step in simplifying algebraic expressions. It involves identifying and extracting common factors from terms. When you have an expression like \(4x^3 - x^3\), notice the common term, which is \(x^3\). You can factor this out:

• The expression becomes \((4 - 1)x^3\).

This process not only simplifies the expression but also reveals underlying patterns that might be useful for additional algebraic manipulations.
In the original exercise, factoring helped in reducing both the cube root and square root expressions to their simplest forms.

Simplifying Expressions

Simplifying algebraic expressions means reducing them to their most basic form without changing the expression's value. It often involves combining like terms after factoring. In the given solutions:

• For the cube root expression, simplifying \((4 - 1)x^3\) resulted in \(3x^3\).
• Similarly, for the square root expression, simplifying \((8 - 1)x^4\) resulted in \(7x^4\).

These simplifications make the expressions easier to interpret and calculate, and are vital for problem-solving in algebra. Always look to reduce expressions to these simplest forms to gain clarity and ensure accuracy.