Problem 112 - Intermediate Algebra

Question

Evaluate \(2 a_{2}^{2}+3 a_{3}^{3}+4 a_{4}^{4}\) for \(a_{2}=2, a_{3}=3,\) and \(a_{4}=4\)

Step-by-Step Solution

Answer

The evaluated expression equals 1113.

Step 1: Plug in the values

Start by substituting the given values for \(a_2, a_3, \) and \(a_4\) into the expression. This gives us: \[2(2)^2 + 3(3)^3 + 4(4)^4\].

Step 2: Evaluate the powers

Evaluate the powers for each term separately. 1. The first term is \(2^2\): \[2^2 = 4\] 2. The second term is \(3^3\): \[3^3 = 27\] 3. The third term is \(4^4\): \[4^4 = 256\].

Step 3: Multiply by coefficients

Multiply each of the results by their respective coefficients:1. For \(2(2)^2\), multiply 2 by 4: \[2 \times 4 = 8\] 2. For \(3(3)^3\), multiply 3 by 27: \[3 \times 27 = 81\] 3. For \(4(4)^4\), multiply 4 by 256: \[4 \times 256 = 1024\].

Step 4: Add the results

Add the products obtained in the previous step to find the final result: \[8 + 81 + 1024 = 1113\].

Key Concepts

Exponents

When dealing with polynomials, exponents are essential for expressing terms that involve powers. An exponent tells you how many times to multiply a number by itself. For example, in the expression \(2^2\), "2" is the base and "2" is the exponent. This means you multiply 2 by itself: \(2 \times 2 = 4\).
Drilling down further, let's consider some simple rules to help understand how to work with exponents:

• Any number raised to the power of 1 is the number itself (e.g., \(a^1 = a\)).
• Any number raised to the power of 0 is 1 (e.g., \(a^0 = 1\), assuming \(a eq 0\)).
• The exponent 2 is called "squared," while the exponent 3 is referred to as "cubed."

These rules form the foundation for evaluating polynomials and understanding how expressions take shape.

Substitution Method

The substitution method is a powerful technique used to evaluate expressions by replacing variables with given values. This is particularly handy in polynomial evaluation, where you substitute variable placeholders with actual numbers.
To apply this method to evaluate the polynomial \(2 a_2^2 + 3 a_3^3 + 4 a_4^4\), follow these steps:

• Identify each variable in the expression: \(a_2\), \(a_3\), and \(a_4\).
• Replace each variable with the provided values: for instance, \(a_2 = 2\), \(a_3 = 3\), and \(a_4 = 4\).
• Plug these values into the expression, transforming it into specific calculations such as \(2(2)^2\), \(3(3)^3\), and \(4(4)^4\).

This method is straightforward and helps simplify expressions to make them easier to solve.

Arithmetic Operations

Once you have substituted the values and evaluated the exponents, the next step involves performing arithmetic operations. These operations include addition, multiplication, and sometimes subtraction or division, applied to the terms of your expression.
In the polynomial \(2 a_2^2 + 3 a_3^3 + 4 a_4^4\), the process involves:

• Calculating each term separately by multiplying the result of the exponent with its coefficient (e.g., \(2 \times 4\) for the first term where \(4\) is \(2^2\)).
• Summing all these results together to obtain the final value. For example, after calculating \(8\), \(81\), and \(1024\), you add them to yield \(1113\).

Essentially, arithmetic operations allow you to piece together individual results into a complete answer, through combining these basic mathematical processes.