Problem 20 - Algebra and Trigonometry

Question

Evaluate the expression using \(x=3, y=4,\) and \(z=-1\). \(\sqrt[4]{x^{3}+14 y+2 z}\)

Step-by-Step Solution

Answer

The expression evaluates to 3.

Step 1: Substitute the Values

Start by substituting the given values of the variables into the expression. The expression is \( \sqrt[4]{x^{3}+14y+2z} \). Substitute \(x=3\), \(y=4\), and \(z=-1\) into the expression: \( \sqrt[4]{3^3+14 \cdot 4+2 \cdot (-1)} \).

Step 2: Calculate Inside the Expression

Now calculate the power of 3, the multiplication, and the addition inside the expression: \(3^3 = 27\) and \(14 \times 4 = 56\). Also compute \(2 \times (-1) = -2\). Substitute these values back into the expression: \( \sqrt[4]{27 + 56 - 2} \).

Step 3: Simplify the Expression

Perform the addition and subtraction inside the square root expression: \(27 + 56 = 83\) and \(83 - 2 = 81\). So, the simplified expression inside the fourth root is \(81\).

Step 4: Evaluate the Fourth Root

Now, evaluate the fourth root of the simplified expression. Since the fourth root of \(81\) is \(3\) (because \(3^4 = 81\)), the expression evaluates to \(3\).

Key Concepts

Substitution in expressions

Substitution in expressions is a fundamental concept in algebra, where we replace variables in an equation or expression with specific values. This process helps simplify and solve expressions.

• Identify the Expression: Look for variables that can be substituted in the expression you are dealing with. For instance, in the expression \( \sqrt[4]{x^{3}+14 y+2 z} \), we have the variables \(x\), \(y\), and \(z\).
• Substitute Values: Replace each variable with the given values. In our example, substitute \(x = 3\), \(y = 4\), and \(z = -1\). This transforms the expression into \( \sqrt[4]{3^3 + 14 \cdot 4 + 2 \cdot (-1)} \).
• Check Your Work: Always double-check that each variable has been correctly substituted to avoid any errors in calculation.

By substituting effectively, you can evaluate and simplify complex mathematical expressions quickly and accurately.

Radical expressions

Radical expressions involve roots, such as square roots or fourth roots, of numbers or algebraic expressions. Understanding and simplifying these expressions require familiarity with root operations.

• Identify the Radical: Look at the radical symbol \( \sqrt[n]{...} \) to understand what type of root you are dealing with. For example, in \( \sqrt[4]{81} \), the number 4 indicates a fourth root.


• Perform Root Evaluation: Calculate the root of the expression inside. To solve \( \sqrt[4]{81} \), we need a number that, when raised to the power of four, equals 81. The number 3 fits this as \(3^4 = 81\).


• Simplifying Radical Expressions: Always try to express the term inside a radical in its simplest form before calculating. This often helps in easy evaluation and verification.

Radicals are an essential part of algebraic manipulation, allowing us to break down larger expressions into simpler forms for easier calculation and understanding.

Exponents and powers

Exponents represent the number of times a base number is multiplied by itself. They are a critical component of math that simplifies calculations and expression evaluation.

• Understand the Base and Exponent: In \(x^3\), \(x\) is the base, and 3 is the exponent, meaning \(x\) is multiplied by itself three times: \(x \cdot x \cdot x\).


• Calculation of Exponents: To evaluate \(3^3\), calculate as \(3 \times 3 \times 3 = 27\). Exponential calculations often form the core steps in simplifying and solving expressions.


• Simplification Techniques: Recognizing patterns and properties of exponents can lessen computational burdens. For example, knowing \((a^m)^n = a^{m \cdot n}\) can streamline operations drastically.

Mastering exponents and powers leads to efficient reasoning in mathematics, enhancing your ability to tackle varied algebraic problems and equations with confidence.