Problem 56 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ 2^{2}\left(10-2^{3}\right) $$

Step-by-Step Solution

Answer

Answer: 8

Step 1: Solve exponents

To begin, we need to solve any exponents in the expression. In our case, we have two: \(2^2\) and \(2^3\). $$ 2^2 = 4 \\ 2^3 = 8 $$ Now, substitute these values back into the expression: $$ 4(10 - 8) $$

Step 2: Solve parentheses

Next, perform the operation inside the parentheses: $$ 10 - 8 = 2 $$ Replace the parentheses with the result: $$ 4(2) $$

Step 3: Solve multiplication

Finally, perform the multiplication: $$ 4 \times 2 = 8 $$ The simplified expression is: $$ 8 $$ Thus, the simplified result of the expression using the order of operations is 8.

Key Concepts

Simplify Expressions

To simplify expressions means to make them as easy to understand as possible. It's like turning a complex recipe into a few simple cooking steps. Just like a chef combines ingredients following a certain order to create a delicious dish, in math, we follow the order of operations to simplify expressions. This involves calculating operations in a specific order: exponents first, then multiplication or division from left to right, and finally addition or subtraction from left to right.

Imagine you're given a math 'recipe' that looks complicated. You start by handling the 'spices'—the exponents. Next, you mix the 'ingredients within bowls'—solve what's inside the parentheses. Then, combine the contents of 'each bowl' – multiplying or dividing. Finally, you 'add the final touches'—doing any addition or subtraction last. By breaking the expression down step by step, you avoid mixing things up, and you'll end up with the correct simplified expression—a 'tasty dish' of a number.

Exponents

Think of exponents like a shorthand in math. Instead of saying 'multiply the number by itself' several times, an exponent tells you exactly how many times to use that number in a multiplication. The number being multiplied is called the 'base', and the exponent tells you the 'height' of your multiplication 'tower'.

For example, in the expression \(2^3\), the base is 2, and the exponent is 3. It's like saying 2 is the building block, and we need to stack it three times: \(2 \times 2 \times 2 = 8\). This 'power' simplifies multiplication and helps you see how quickly numbers can grow. Remember to handle these towering numbers first when simplifying expressions; it’s like laying the foundation of a building before adding more parts.

Arithmetic Operations

Arithmetic operations are the building blocks of most mathematical equations, much like the basic commands we give a computer. The primary operations include addition (+), subtraction (-), multiplication (\(\times\)), and division (\(\div\)).

It's crucial to follow the right order, like putting on socks before shoes. This order is usually abbreviated as PEMDAS: parentheses first, exponents next, multiplication and division (from left to right), and addition and subtraction last (from left to right).

An understanding of how to use these operations together correctly ensures you can simplify expressions down to their most straightforward form. It empowers you to tackle more complex problems, like layering more commands to build complicated computer programs or putting together a multi-step DIY project – each part completed in sequence to build the whole.