Problem 90 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{3^{2}+4}{5}\)

Step-by-Step Solution

Answer

The value of the expression is 2.6.

Step 1: Evaluate the Exponent

First, calculate the value of the exponent in the expression. The expression contains \(3^2\), which means 3 multiplied by itself. Thus, \(3^2 = 3 \times 3 = 9\).

Step 2: Add the Result to the Next Number

Add the result from Step 1 to 4, as indicated in the expression. This means adding 9 and 4 together: \(9 + 4 = 13\).

Step 3: Divide by 5

Finally, divide the result from Step 2 by 5. This means calculating \(\frac{13}{5}\). When dividing 13 by 5, you get \(2.6\) or \(2 \frac{3}{5}\).

Key Concepts

Exponentiation

Exponentiation in mathematics is a vital concept that appears often in algebraic expressions. It involves raising a number, known as the base, to the power of an exponent. In simple terms, it means multiplying the base by itself a certain number of times. For example, in the expression \(3^2\), the base is 3, and the exponent is 2. This means you multiply 3 by itself: \[3^2 = 3 \times 3 = 9\] Exponentiation is a fundamental operation, much like addition or multiplication, and it's crucial for simplifying algebraic expressions. Remember, the exponent tells you how many times you need to multiply the base by itself.

• Base: The number you are multiplying. In \(3^2\), the base is 3.
• Exponent: The number of times you multiply the base by itself. In \(3^2\), the exponent is 2.
• Result: The product of the base multiplied the indicated number of times, which for \(3^2\) is 9.

Practicing exponentiation helps you develop a solid understanding of how numbers grow exponentially and is essential for more advanced math topics.

Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics and include addition, subtraction, multiplication, and division. In this particular problem, we are dealing with addition and exponents within the context of an algebraic expression. After evaluating the exponent \(3^2 = 9\), the next step in the expression \(\frac{3^{2}+4}{5}\) is to perform an addition operation. This means you'll take the value obtained from the exponentiation and add 4: \[9 + 4 = 13\] This step highlights how arithmetic operations are applied in sequence following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication, Division, Addition, and Subtraction. It's important to apply these rules to solve any algebraic expression accurately. Understanding arithmetic operations ensures that you accurately manipulate numbers and maintain the integrity of mathematical expressions.

• Addition: The arithmetic process of bringing two numbers together to form a new total. Here, 9 and 4 are added to get 13.
• Order of Operations: A set of rules that dictates the sequence in which operations should be performed to accurately solve an expression.

Fraction Division

Fraction division involves dividing the numerator by the denominator to find the quotient. In the expression \(\frac{3^{2}+4}{5}\), after you have completed all previous steps, you need to divide the sum 13 by 5. This is the final step: \[\frac{13}{5} = 2.6 \] When dividing fractions, you're actually distributing the numerator value across the denominator. This process provides either a decimal or a mixed number. For example: \[ \frac{13}{5} = 2 \frac{3}{5} \] Understanding how to handle division in fractions is crucial since it comes up often when solving algebraic expressions. Practice with fraction division is essential to developing a fluency in mathematics, enabling you to tackle more complex algebraic problems. Here are some basics to keep in mind:

• Numerator: The top number in a fraction. It signifies the amount to be divided.
• Denominator: The bottom number in a fraction. It indicates the number of equal parts the whole is divided into.
• Quotient: The result obtained from the division of the numerator by the denominator.
• Mixed Number: A quotient expressed as a whole number alongside a fraction (if any reminder is left), like \(2 \frac{3}{5}\).