Problem 40 - Algebra 1

Question

Evaluate the expression. $$4 a^{2}+11 \text { when } a=5$$

Step-by-Step Solution

Answer

The evaluated expression when \(a = 5\) is 111.

Step 1: Substitute Value

Substitute the given value of \(a = 5\) into the expression \(4a^{2}+11\). This gives us the expression \(4*(5)^{2}+11\).

Step 2: Apply Exponent

In the expression from step 1, apply the exponent first according to the order of operations (PEMDAS/BODMAS). This gives us \(4*25 + 11\).

Step 3: Perform Multiplication

Next, perform the multiplication operation. \(4*25 = 100\). So the expression becomes \(100 + 11\).

Step 4: Perform Addition

Finally, perform the addition operation to finalize the solution. \(100 + 11 = 111\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating algebraic expressions. It's the protocol that guides us on which operations to perform first to ensure consistency in our results. The acronym PEMDAS is often used in the US to remember the order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), and it's known as BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) elsewhere.

For example, when you come across an expression like \(4 a^{2} + 11\) with \(a = 5\), following PEMDAS will result in a correct evaluation. Mathematically, it translates to:

Exponents in Algebra

Exponents play a pivotal role in algebra and can drastically change the value of an expression. An exponent represents how many times a number, known as the base, is multiplied by itself. For instance, \(5^2\) means that 5 is multiplied by itself once: \(5 * 5 = 25\).

Understanding exponents is essential for evaluating expressions correctly. If you encounter an expression with an exponent, like \(4a^2\), calculate the exponent before other operations if no parentheses indicate otherwise. The correct approach for the example where \(a = 5\) gives us \(4*5^2\), which is calculated as \(4*25\) after the exponent is applied.

Substitution in Algebra

Substitution is a fundamental technique used in algebra to evaluate expressions with variables. When an expression contains a variable, you can 'substitute' it with its given value to find the numerical result. For example, if you're given \(4a^2 + 11\) and told that \(a = 5\), you replace every instance of \(a\) with 5, which results in \(4*(5)^2 + 11\).

This direct replacement makes solving the expression straightforward. Substitution isn’t just limited to numbers; it can involve replacing variables with other expressions or functions, depending on what you’re trying to solve.