Problem 1 - College Algebra

Question

Evaluate each exponential expression. $$ 5^{2} \cdot 2 $$

Step-by-Step Solution

Answer

The result of the expression \(5^{2} \cdot 2\) is 50.

Step 1: Evaluate the Exponential Expression

When working with exponents, the exponential expression should be evaluated first. This operation is denoted by \(5^2\) which means 5 multiplied by itself. This operation will therefore yield 25.

Step 2: Perform the Multiplication

After the exponent is evaluated, multiply the result by the constant, thus \(25*2=50\)

Key Concepts

Evaluate Exponents

To evaluate exponents means to calculate the value of a number raised to a certain power. The power, also known as the exponent, indicates how many times the base number is to be multiplied by itself. For example, in the expression \(5^2\), the number 5 is the base and 2 is the exponent. Evaluating \(5^2\) involves multiplying 5 by itself once because the exponent 2 signifies two instances of the base: \(5 \times 5 = 25\).

It's essential to handle the exponent before any other operations when it appears in a more complex mathematical expression, such as \(5^2 \cdot 2\). The exponent modifies the base directly, and its result will then interact with the other numbers or operations present in the expression.

Exponential Notation

Exponential notation is a convenient way of expressing repeated multiplication of the same factor. It consists of two parts: the base and the exponent. The base is the number being multiplied, and the exponent, typically written as a superscript, tells us how many times the base is used as a factor in the multiplication. So, if we have \(a^n\), 'a' is the base and 'n' is the exponent. An essential property of the exponential notation is that any number to the power of 1 is itself (\(a^1 = a\)) and any number to the power of 0 is 1 (\(a^0 = 1\)), with the exception that 0 to the power of 0 is undefined.

Understanding this notation is crucial because it's used widely in mathematics, from representing large or small numbers to expressing the growth of investments or populations.

Multiplication with Exponents

Multiplication with exponents comes into play when an exponential expression is to be multiplied by another number or a different exponential expression with the same base. When the base is the same, the exponents can be added together (e.g., \(a^n \times a^m = a^{n+m}\)). However, when the exponential expression is multiplied by a constant, as in our exercise \(5^2 \cdot 2\), you evaluate the exponent first and then multiply the result by the constant. So, after calculating \(5^2 = 25\), we multiply that result by 2 to get \(25 \cdot 2 = 50\).

This principle is part of the Order of Operations, which dictates that exponentiation should be addressed before multiplication or division unless groupings like parentheses dictate otherwise.