Problem 95 - Algebra A Combined Function

Question

Simplify each expression. $$ 2^{3}+2^{0} $$

Step-by-Step Solution

Answer

The expression simplifies to 9.

Step 1: Evaluate the Powers

First, evaluate each power individually. The expression given is: \[ 2^3 + 2^0 \] Calculate each power: \[ 2^3 = 2 imes 2 imes 2 = 8 \] \[ 2^0 = 1 \] because any non-zero number raised to the power of 0 is 1.

Step 2: Add the Evaluated Powers

Now, add the evaluated powers from Step 1.Combine the results: \[ 8 + 1 = 9 \].

Key Concepts

Powers of numbers

Powers of numbers, also known as exponents, are a way to express repeated multiplication of a number. When we see something like "\(2^3\)", it means we multiply 2 by itself 3 times. This is written as \(2 \times 2 \times 2\) and equals 8. The number "2" here is called the base, and "3" is the exponent or power.

Exponents make math easier by simplifying the representation of large calculations. Instead of writing many multiplications, we can easily write an expression like \(5^{4}\) to quickly understand it means \(5 \times 5 \times 5 \times 5\).

Remember, there are specific rules for exponents:

• Any number raised to the power of 1 is the number itself, for example, \(7^1 = 7\).
• Any non-zero number raised to the power of 0 is always 1, such as \(3^0 = 1\).

Understanding these basic rules will help you handle more complex expressions.

Order of operations

When simplifying mathematical expressions, the order of operations is crucial. This rule dictates the sequence in which operations should be performed to ensure accurate results. The standard process can be remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.

With our example, \(2^3 + 2^0\), we first handle the exponents before moving on to any additions or subtractions. In following the order of operations, we:

• Start with any calculations inside Parentheses, if they exist.
• Calculate the Exponents next.
• Proceed with Multiplication and Division, from left to right.
• Finally, address Addition and Subtraction, from left to right.

This structured approach helps eliminate mistakes and provides a clear path to simplifying expressions correctly.

Simplifying expressions

Simplifying expressions involves breaking down complex calculations into simpler ones. It often requires following the order of operations correctly to reach the simplest form of the expression.

In our expression \(2^3 + 2^0\), to simplify it, we first compute the powers. We find that \(2^3 = 8\) and \(2^0 = 1\). By adding these results, we obtain 9. Therefore, the simplified version of the initial expression is simply 9.

When simplifying expressions:

• Perform calculations in a step-by-step fashion, focusing on one operation at a time.
• Reassess your work at each stage to ensure no errors have occurred.
• Keep practicing, as familiarity makes simplification more intuitive and automatic over time.

This methodical approach helps students become more adept at handling various mathematical problems effectively.