Evaluating Algebraic Expressions with Exponents

The question asks us to find the value of the expression \( 2x^3 - 6x - 5 \), given that \( x = 2^{1/3} + 2^{-1/3} \).

Let's break down the problem step-by-step.

Step 1: Understand the Given Value of x

We are given the value of \( x \) as a sum of two terms involving fractional exponents:

\( x = 2^{1/3} + 2^{-1/3} \)

This expression looks like it could simplify nicely when cubed.

Step 2: Cube the Expression for x

To evaluate \( 2x^3 - 6x - 5 \), we first need to find \( x^3 \). We can cube both sides of the equation for \( x \).

\( x^3 = (2^{1/3} + 2^{-1/3})^3 \)

We use the algebraic identity for cubing a sum: \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \).

Let \( a = 2^{1/3} \) and \( b = 2^{-1/3} \).

Then,

• \( a^3 = (2^{1/3})^3 = 2^{(1/3) \times 3} = 2^1 = 2 \)
• \( b^3 = (2^{-1/3})^3 = 2^{(-1/3) \times 3} = 2^{-1} = \frac{1}{2} \)
• \( ab = (2^{1/3})(2^{-1/3}) = 2^{1/3 + (-1/3)} = 2^{1/3 - 1/3} = 2^0 = 1 \)
• \( a+b = 2^{1/3} + 2^{-1/3} = x \) (from the original definition of x)

Substitute these values back into the identity \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \):

\( x^3 = 2 + \frac{1}{2} + 3(1)(x) \)

\( x^3 = 2 + 0.5 + 3x \)

\( x^3 = 2.5 + 3x \)

We can also write 2.5 as \( \frac{5}{2} \):

\( x^3 = \frac{5}{2} + 3x \)

Step 3: Substitute x³ into the Target Expression

Now we substitute the expression for \( x^3 \) we found into the expression \( 2x^3 - 6x - 5 \).

The target expression is \( 2x^3 - 6x - 5 \).

Substitute \( x^3 = \frac{5}{2} + 3x \):

\( 2\left(\frac{5}{2} + 3x\right) - 6x - 5 \)

Step 4: Simplify the Expression

Now, distribute the 2 and combine like terms:

\( 2 \times \frac{5}{2} + 2 \times 3x - 6x - 5 \)

\( 5 + 6x - 6x - 5 \)

Combine the terms with \( x \):

\( (6x - 6x) + (5 - 5) \)

\( 0 + 0 \)

\( 0 \)

The value of the expression \( 2x^3 - 6x - 5 \) is 0.

Summary of Steps

Here is a quick overview of the process:

1. Identify the given value of \( x \).
2. Cube \( x \) using the formula \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \), substituting \( a=2^{1/3} \) and \( b=2^{-1/3} \).
3. Simplify the expression for \( x^3 \).
4. Substitute the simplified \( x^3 \) into the expression \( 2x^3 - 6x - 5 \).
5. Simplify the resulting expression to find the final value.

Let's put the key intermediate result in a table.

The final calculation yielded 0.

Revision Table - Key Concepts

Understanding the properties of exponents and algebraic identities is crucial for solving this type of problem.

Additional Information - Related Problems

Problems involving expressions like \( x = a^{1/3} + a^{-1/3} \) often simplify nicely when finding expressions involving \( x^3 \). The key is the identity \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \), where \(ab\) often simplifies to a constant when \( a \) and \( b \) are reciprocals with roots, like \( a^{1/n} \) and \( a^{-1/n} \).

Consider a general case: If \( x = a^{1/3} + a^{-1/3} \), then

\( x^3 = (a^{1/3} + a^{-1/3})^3 = (a^{1/3})^3 + (a^{-1/3})^3 + 3(a^{1/3})(a^{-1/3})(a^{1/3} + a^{-1/3}) \)

\( x^3 = a + a^{-1} + 3(a^0)(x) \)

\( x^3 = a + \frac{1}{a} + 3x \)

\( x^3 - 3x = a + \frac{1}{a} \)

In our specific problem, \( a=2 \). So \( x = 2^{1/3} + 2^{-1/3} \).

Then \( x^3 - 3x = 2 + \frac{1}{2} = \frac{5}{2} \).

From \( x^3 = \frac{5}{2} + 3x \), we have \( x^3 - 3x - \frac{5}{2} = 0 \).

The expression we needed to evaluate was \( 2x^3 - 6x - 5 \). We can factor out a 2:

\( 2(x^3 - 3x - \frac{5}{2}) \)

Since \( x^3 - 3x - \frac{5}{2} = 0 \), the expression becomes \( 2(0) = 0 \).

This confirms our earlier calculation and shows a common pattern for such problems.