Problem 49 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lessons 1.2,1.3) $$ 2^{6}-3+1 $$

Step-by-Step Solution

Answer

The evaluated expression results in 62.

Step 1: Calculate Exponent

Calculate the value of \(2^6\). Exponentiation is done first according to BIDMAS/BODMAS rules.

Step 2: Subtract

After calculation, \(2^6\) equals 64. Now subtract 3 from 64. Subtraction is done second following the BIDMAS/BODMAS rules.

Step 3: Addition

After subtraction, the result is 61. Add 1 to 61. Addition is the last operation to be done according to BIDMAS/BODMAS rules.

Key Concepts

Exponentiation

Exponentiation might seem tricky at first, but it is just repeated multiplication. When you see something like \(2^6\), it means you should multiply 2 by itself 6 times: \[2 \times 2 \times 2 \times 2 \times 2 \times 2\]This equals 64.
It's important to remember that exponentiation is done first in the order of operations, often remembered by the acronym BIDMAS or BODMAS—where 'E' in some variations stands for Exponents.

• Exponentiation — Done before other operations like addition or subtraction.
• Power — Represents how many times the base is multiplied by itself.

So, whenever you encounter an exponent, handle it before moving on to other mathematical operations. In our example, we handled \(2^6\) first to get 64 before doing anything else.

Subtraction

Subtraction involves taking one number away from another. Once you have calculated any exponentiation present, subtraction usually comes next in the order of operations. In the exercise, after calculating \(2^6\) to obtain 64, we then move to subtract 3.
This operation reduces that number to 61, because:\[64 - 3 = 61\]

• Subtraction — You remove a specific number of units from the total.
• Next step — Comes immediately after exponentiation when processing a mathematical expression.

Remember: While subtraction is straightforward, it's vital to remember the order it is performed in. Always deal with subtraction in sequence, after handling any exponentiation issues. This ensures you maintain accurate results throughout the calculation process.

Addition

Addition is one of the most fundamental operations in mathematics. Once you've completed any necessary subtractions, addition follows as the final arithmetic operation. In our equation, after resolving the subtraction to get 61, we then add 1:\[61 + 1 = 62\]

• Addition — Used to combine numbers into a larger total.
• Order adherence — Completed after other high-priority operations like exponentiation and subtraction.

It's always performed last when considering the traditional order of operations. However, it's critical to always follow BIDMAS/BODMAS to ensure nothing is missed. By doing this in our example, we ensure the final answer is confidently 62.