Problem 13 - Algebra 1

Question

Evaluate the expression when \(x=3\) \(5^{x}\)

Step-by-Step Solution

Answer

The evaluated result of the expression when \(x=3\) is 125.

Step 1: Substitution

Substitute the value \(x=3\) into the expression \(5^{x}\). This results in \(5^{3}\).

Step 2: Simplify

The exponential operation \(5^{3}\) means 5 multiplied by itself 3 times. So, this simplifies to \(5*5*5\).

Step 3: Calculate

Upon performing the multiplication, we get 5*5*5 = 125. Thus, the expression \(5^{3}\) simplifies to 125.

Key Concepts

Evaluating Expressions

Evaluating mathematical expressions is a fundamental skill that involves replacing variables with their corresponding values and simplifying the result using the order of operations. For example, when given the expression \( 5^{x} \) and instructed to evaluate it at \( x=3 \), it's important to understand that evaluating means finding the expression's numerical value given that \( x \) has a specific value.

Evaluation begins with the substitution method, where the given value replaces the variable within the expression. Here with \( x=3 \) substituted, the problem simplifies to \( 5^{3} \). The next step is applying the rules of exponentiation to simplify further. This skill is vital for students learning algebra, as it lays the groundwork for solving more complex equations and functions. One improvement advice for this step is demonstrating substitution with varied examples, therefore reinforcing the concept. Moreover, explaining that evaluating expressions can apply to any mathematical operation—not just exponentiation—helps establish a broader understanding of the concept.

Exponentiation

Exponentiation is the mathematical operation, involving two numbers, the base (\( a \) and the exponent (\( n \) which is also called the power. The operation \( a^{n} \) denotes the base \( a \) being multiplied by itself \( n \) times. For example, \( 5^{3} \) involves multiplying 5 by itself three times: \( 5*5*5 \).

Understanding exponentiation is essential for various areas of mathematics and science. It's a building block for more advanced topics like exponential growth, decay, and compound interest calculations.

Common Mistakes

Common pitfalls include confusing the terms 'base' and 'exponent' or misapplying the exponent to only the first term when the base is a product or fraction. Students should practice this concept with diverse bases, including negative numbers and fractions, to build competency. It's advisable to engage with exercises that include both numerical and algebraic bases for better depth in understanding exponentiation.

Substitution Method

The substitution method is a technique used to simplify expressions or solve equations that involve variables. The method includes replacing a variable with a given number or another expression. Substitution plays a critical role when evaluating expressions or solving systems of equations.

The steps involved in the substitution method begin with having a clear understanding of which variable is being replaced. As seen in our example with \( 5^{x} \), \( x \) was substituted with 3 before applying exponentiation.

Why Use Substitution?

This method is especially useful in algebra to reduce complex relations, evaluate functions for specific inputs, and solve equations where one variable can be expressed in terms of another. Encouraging students to check their work after substitution helps in avoiding errors and ensures understanding of the relationships between the variables. To improve this method's understanding, introducing real-world scenarios where substitution is applicable could make the concept more relatable and solidify the student's grasp of its applications.