Problem 60 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ 1^{6}+0^{8}+5^{2}(2+8)^{3} $$

Step-by-Step Solution

Answer

Answer: The simplified result of the expression is $25001$.

Step 1: Rewrite the Expression

Begin by rewriting the expression given in the exercise: $$ 1^{6}+0^{8}+5^{2}(2+8)^{3} $$

Step 2: Parentheses/Brackets

Start by evaluating the expression inside the parentheses: $$ (2+8)=10 $$ Now rewrite the expression with the evaluated parentheses: $$ 1^{6}+0^{8}+5^{2}(10)^{3} $$

Step 3: Exponents/Orders

Next, evaluate the powers in the expression: $$ 1^{6}=1 \\ 0^{8}=0 \\ 5^{2}=25 \\ (10)^{3}=1000 $$ Now rewrite the expression with the evaluated powers: $$ 1+0+25\cdot 1000 $$

Step 4: Multiplication and Division

In this expression, there is only one multiplication operation to perform. We have: $$ 25\cdot 1000 = 25000 $$ Rewrite the expression with the multiplication completed: $$ 1+0+25000 $$

Step 5: Addition and Subtraction

Finally, we will perform the addition operations in the expression: $$ 1+0+25000 = 25001 $$ The expression has been simplified to the final result as: $$ 1^{6}+0^{8}+5^{2}(2+8)^{3} = 25001 $$

Key Concepts

Simplify Mathematical Expressions

When we simplify mathematical expressions, it's like we're decoding a puzzle, using specific rules to find out what the expression is really saying. Simplifying is crucial because it helps us find an expression's most straightforward form, making it easier to work with or understand. Especially when dealing with complex expressions, breaking them down step-by-step can prevent errors and confusion. For instance, the example from our textbook exercise starts with an expression that looks complicated, but with each step of simplification, we peel away layers until we get a single number which is our answer. Remember, simplifying isn't just about getting to an answer—it's about understanding the journey there.

Exponentiation

Just as superheroes have special powers, numbers can have powers too—this is what we call exponentiation. It's a way of saying that a number is multiplied by itself a certain number of times. In the mathematical world, we write this with a little number high up to the right of our main number, like this: \(5^3\), which says '5 to the power of 3'. This means 5 multiplied by itself 3 times (5 x 5 x 5). Don’t be fooled by appearances; even exponents that seem weak—like \(1^6\) or \(0^8\) from our textbook problem—are simply following the same rules. No matter what the exponent is, we have to calculate these 'powers' to simplify expressions correctly.

Parentheses in Math

In math, we sometimes use parentheses, ( ), to group numbers together. Think of them like a hug around certain parts of a math problem—they’re telling us, 'Hey! Calculate what's inside first before dealing with anything outside!' This is exactly what we do in our textbook exercise; the number bond inside the parentheses is so strong that we have to respect it and do that calculation before anything else. Just like in a conversation, where we might say something important in parentheses for emphasis, in math, these not-so-silent symbols help us focus on what's important first.

Arithmetic Operations

Arithmetic operations are the bread and butter of math—the basic actions we can perform with numbers. They're like the tools in a toolbox that let us put together or take apart numbers in different ways. There are four main types: addition \(+\), subtraction \(-\), multiplication \(\times\), and division \(\div\). But here's the twist: we can't just use these tools any way we like. We have to follow a specific order, called the order of operations, to get the job right without any mix-ups. This order is especially salient when we work through problems like in our exercise, where each operation we perform brings us closer to that one final, correct answer.