Problem 26 - Precalculus Mathematics for Calculus

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

Substitute the given values of the variables into the expression. Substitute \(x = 3\), \(y = 4\), and \(z = -1\) into \(x^3 + 14y + 2z\). This gives us: \((3)^3 + 14(4) + 2(-1)\).

Step 2: Calculate the expression inside the fourth root

Now, calculate \((3)^3 + 14(4) + 2(-1)\). First, compute each separate term: \(3^3 = 27\), \(14 \times 4 = 56\), and \(2 \times (-1) = -2\). Then add them together: \(27 + 56 - 2 = 81\).

Step 3: Evaluate the fourth root

Now that we have the expression inside the fourth root calculated as 81, find the fourth root of 81. The fourth root of 81 is 3, because \(3^4 = 81\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of the same number by itself. In our exercise, we see the term \(x^3\). Here, the exponent is 3, which means we multiply \(x\) by itself three times. If we substitute \(x\) with 3, the calculation becomes \(3 \times 3 \times 3 = 27\).
Understanding how to handle exponents is crucial because they appear often in algebra and higher-level math.
The base, \(x\) in our example, is the number being multiplied. The exponent tells us how many times to use that base in a multiplication.

• An exponent of 2 (\(x^2\)) is often called "squared."
• An exponent of 3 (\(x^3\)) is referred to as "cubed."

Grasping exponents allows you to simplify expressions and solve more complex equations.

Substitution

Substitution is a method used in algebra to replace variables with their respective numerical values.
This technique makes it possible to simplify or evaluate expressions.
In our exercise, we are given specific values: \(x=3\), \(y=4\), and \(z=-1\). Substituting these into the expression \(x^3 + 14y + 2z\), we need to "plug in" the numbers where the variables are, giving us \((3)^3 + 14(4) + 2(-1)\).This method is especially useful when solving:

• Algebraic expressions that involve multiple variables.
• Equations where particular values of the variables are provided.

By substituting, you are converting the expression into numerical form, which makes the final computations straightforward.

Fourth Root

A fourth root is a type of radical expression where you look for a number that, when multiplied by itself four times, equals the given expression.
In our problem, we find the fourth root of 81.
This means we need a number that, when raised to the power of four, equals 81. After evaluating \(81 = 3^4\), you see that the fourth root of 81 is 3.Understanding how to compute a fourth root is important:

• The operation is opposite of taking the exponent; it "undoes" the power of four.
• It helps in solving equations where a number is raised to the fourth power.

Like square roots and cube roots, knowing how to handle fourth roots expands your ability to work with higher-degree equations.