Problem 27 - Precalculus Mathematics for Calculus

Question

Evaluate the expression using \(x=3, y=4,\) and \(z=-1\). $$(9 x)^{2 / 3}+(2 y)^{2 / 3}+z^{2 / 3}$$

Step-by-Step Solution

Answer

The expression evaluates to 14.

Step 1: Substitute values into the expression

Insert the provided values for the variables into the expression: substitute \(x = 3\), \(y = 4\), and \(z = -1\) into \[(9x)^{2/3} + (2y)^{2/3} + z^{2/3}.\] This results in \[(9 \times 3)^{2/3} + (2 \times 4)^{2/3} + (-1)^{2/3}.\]

Step 2: Simplify each term

Calculate each term within the expression: 1. \(9 \times 3 = 27\), resulting in \(27^{2/3}\).2. \(2 \times 4 = 8\), resulting in \(8^{2/3}\).3. \((-1)^{2/3}\) is simply calculated.

Step 3: Compute each term’s power

Find the \(2/3\) power of each term: 1. \(27^{2/3}:\) The cube root of 27 is 3. Then, \(3^2 = 9\).2. \(8^{2/3}:\) The cube root of 8 is 2. Then, \(2^2 = 4\).3. \((-1)^{2/3}:\) The cube root of -1 is -1. Then, \((-1)^2 = 1\).

Step 4: Sum the results

Add the values obtained: \[9 + 4 + 1 = 14.\]

Key Concepts

Exponentiation

Exponentiation involves raising a base number to a given power or exponent. In this exercise, we work with fractional exponents, specifically the exponent \( \frac{2}{3} \).
This means that we are essentially performing two mathematical operations: first finding the cube root and then squaring the result.A fractional exponent like \( a^{\frac{2}{3}} \) can be understood as:

• Cube Root: Find the cube root of \( a \).
• Square: Raise the result to the power of 2.

Using these steps helps simplify the evaluation of expressions with fractional exponents.

Substitution

Substitution is the process of replacing variables in an expression with known values. To evaluate an expression, we substitute the given values into the expression, making it numerically solvable.
In our example:

• Replace \( x \) with 3.
• Replace \( y \) with 4.
• Replace \( z \) with -1.

After substitution, the expression \((9x)^{\frac{2}{3}} + (2y)^{\frac{2}{3}} + z^{\frac{2}{3}}\) becomes \((27)^{\frac{2}{3}} + (8)^{\frac{2}{3}} + (-1)^{\frac{2}{3}}\).
Substitution simplifies complex algebraic expressions into straightforward numerical calculations.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest form while retaining the same value. This often involves combining like terms or removing parentheses by performing indicated operations.
For this task, we simplified each term within the expression:

• The term \(9x\) simplifies to \(27\).
• The term \(2y\) simplifies to \(8\).
• The term \(z\) remains \(-1\).

This simplification makes the subsequent steps of evaluating exponentiation easier and more manageable.

Cube Roots

Understanding cube roots, or the third root of a number, is essential when dealing with fractional exponents. The cube root of a number \( a \) is a value \( b \) such that \( b^3 = a \).
In this particular problem, when calculating \( (27)^{\frac{2}{3}} \), first find the cube root of 27, which is 3, and then square it.
Similarly, for \( (8)^{\frac{2}{3}} \), the cube root of 8 is 2, leading to further analysis. And \( (-1)^{\frac{2}{3}} \) gives -1 which then is squared.

• The cube root of 27 is 3.
• The cube root of 8 is 2.
• The cube root of -1 is -1.

These calculated cube roots are then raised to the power of 2 to complete the evaluation of the expression.