Problem 71 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. \(5^{2} \cdot 5^{1}\)

Step-by-Step Solution

Answer

The answer to the expression \(5^{2} \cdot 5^{1}\) is 125.

Step 1: Identify the base numbers and their powers

In this expression \(5^{2} \cdot 5^{1}\) the base number is 5. The number 5 is raised to the power of 2 in the first part and to the power of 1 in the second part.

Step 2: Apply the laws of exponents

By the laws of exponents, when multiplying, if the base numbers are the same, then add the exponents. So, \(5^{2} \cdot 5^{1}=5^{2+1}=5^{3}\)

Step 3: Simplify the result

Now that the exponents are added up, calculate the power of 5. \(5^{3}=125\)

Key Concepts

Evaluate Expressions

When we speak of evaluating expressions in mathematics, we refer to the process of simplifying or finding the value of an expression based on the rules of mathematical operations and the numbers involved. Evaluating an exponential expression involves applying specific knowledge of how exponentiation works.

For instance, take the expression given in our example, which is to evaluate the expression involving exponential numbers: it's essential to deal with the individual components correctly. The expression given is simple, consisting of two terms that have the same base. To evaluate such expressions, you apply the laws of exponents, which provide a structured approach to reach the expression’s simplest form or value.

Understanding how to evaluate expressions makes more complicated mathematics much more accessible as it forms the foundation for algebra, calculus, and beyond.

Exponentiation

Exponentiation is a form of mathematical shorthand for expressing repeated multiplication of the same number. The number being multiplied, known as the base, is written once followed by a superscript number indicating how many times it should be multiplied by itself. For instance, in the expression \(5^{2}\), the base is 5, and the exponent is 2, which means 5 is multiplied by itself once: \(5 \times 5\).

Understanding exponentiation requires familiarization with several key terms:

• Base: The number that is being multiplied by itself.
• Exponent: Indicates how many times the base is used as a factor in the multiplication.
• Power: The entire expression, such as \(5^{2}\), which can also be called '5 raised to the 2nd power'.

This concept is essential when working with larger numbers, in fields ranging from engineering and computer science to finance and sciences, as it provides a way to express and work with very large or very small numbers efficiently.

Simplifying Expressions

Simplifying expressions is a cornerstone of algebra. This process involves reducing an expression to its most basic form without changing its value. Simplification makes equations easier to interpret and solve. One way to simplify an expression is to combine like terms - terms that have the same variables raised to the same power.

Using the laws of exponents is a vital part of this simplification process. For example, in our exercise, two terms with the same base but different exponents are multiplied. According to the laws, you add the exponents when the bases are the same: \(5^{2} \times 5^{1} = 5^{2+1} = 5^{3}\). This simplifies the expression from two separate multiplicative components to a single power of 5.

Simplifying expressions is a skill that will be used repeatedly in various mathematical contexts, so mastering the laws of exponents provides a valuable tool for both academic and real-world applications in quantitative reasoning.