Unraveling Complex Math: A Step-by-Step Guide

Hey math enthusiasts! Ready to dive into some intriguing math problems? In this guide, we'll break down the expressions:

• $b \left(5^7+5^6+5^5\right): 5^5+9^7: 3^{10}$
• $3^3-3^2:\left(2^6-2^5-2^4-2^3-2^2-2^1-2^0\right)$
• $2 \cdot 2^2 \cdot 2^{2^2}:\left[2+2^2+2^{2^2}:\left(2^{2^2}-2^2 \cdot 2\right)\right]$

We'll tackle these step by step, making sure everything is clear and easy to follow. Let's get started!

Deciphering the First Expression: $\bf{b \left(5^7+5^6+5^5\right): 5^5+9^7: 3^{10}}$

Alright, let's start with the first expression: $b \left(5^7+5^6+5^5\right): 5^5+9^7: 3^{10}$. First, we need to clarify what the variable 'b' represents within the context of this mathematical expression. Since no specific value is provided for 'b', we will assume it is a constant multiplier unless otherwise specified. We can simplify this expression, applying the order of operations, which is crucial for solving such problems correctly. Remember, the order of operations is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Let's go through the simplification process.

Step-by-Step Simplification

1. Factor out the common term: Inside the parentheses, we see $5^7$, $5^6$, and $5^5$. We can factor out $5^5$ because it's the smallest power of 5 present in all three terms. This gives us: $b \cdot 5^5 \cdot (5^2 + 5^1 + 1)$. This step is essential because it simplifies the calculation significantly.
2. Calculate the terms inside the parentheses: $5^2 = 25$, $5^1 = 5$, and $1$ remains as it is. So, we have $25 + 5 + 1 = 31$. Our expression now looks like this: $b \cdot 5^5 \cdot 31: 5^5+9^7: 3^{10}$.
3. Simplify the division of powers of 5: Now, we have $b \cdot 5^5 \cdot 31: 5^5$. Since we are dividing $b \cdot 5^5$ by $5^5$, these terms cancel each other out, leaving us with: $b \cdot 31$. So the expression is now: $b \cdot 31 + 9^7: 3^{10}$.
4. Simplify the exponents and division involving 9 and 3: We know that $9 = 3^2$, so $9^7 = (3^2)^7 = 3^{14}$. The expression now is $b \cdot 31 + 3^{14} : 3^{10}$. When dividing exponents with the same base, you subtract the powers, thus $3^{14} : 3^{10} = 3^{14-10} = 3^4$. Our expression simplifies to: $b \cdot 31 + 3^4$.
5. Calculate the final exponent: $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$.
6. Final simplified form: The final simplified form of the expression is $b \cdot 31 + 81$. If 'b' is a known constant, you would then substitute its value and calculate the final result. In the absence of a value for 'b', the expression can be left as $31b + 81$.

In essence, we've carefully broken down the initial expression into manageable steps, applying the order of operations and properties of exponents to arrive at the simplified form. This demonstrates how complex mathematical problems can be systematically solved by understanding fundamental principles.

Tackling the Second Expression: $\bf{3^3-3^2:\left(2^6-2^5-2^4-2^3-2^2-2^1-2^0\right)}$

Now, let's move on to the second expression: $3^3-3^2:\left(2^6-2^5-2^4-2^3-2^2-2^1-2^0\right)$. Remember, the order of operations is our guiding light here. We'll start by simplifying the terms inside the parentheses first. Within the parentheses, we have a series of exponents involving the base 2. This part requires careful calculation of each power and subtraction.

Detailed Breakdown

1. Calculate the powers of 2: We have $2^6 = 64$, $2^5 = 32$, $2^4 = 16$, $2^3 = 8$, $2^2 = 4$, $2^1 = 2$, and $2^0 = 1$. It’s essential to calculate each exponent accurately.
2. Perform the subtraction inside the parentheses: Now, we subtract all the terms: $64 - 32 - 16 - 8 - 4 - 2 - 1$. Doing this step by step, we get $64 - 32 = 32$, $32 - 16 = 16$, $16 - 8 = 8$, $8 - 4 = 4$, $4 - 2 = 2$, and finally, $2 - 1 = 1$. Therefore, the expression inside the parentheses simplifies to 1.
3. Simplify the remaining terms: We now have $3^3 - 3^2 : 1$. Calculate $3^3 = 27$ and $3^2 = 9$. So the expression is $27 - 9:1$.
4. Perform the division: The expression becomes $27 - 9 = 18$ since anything divided by 1 is itself.
5. Final answer: Therefore, the simplified form of the entire expression is 18. This result shows how understanding and meticulously applying the order of operations, step by step, helps us solve mathematical problems.

Deconstructing the Third Expression: $\bf{2 \cdot 2^2 \cdot 2^{2^2}:\left[2+2^2+2^{2^2}:\left(2^{2^2}-2^2 \cdot 2\right)\right]}$

Let's move on to the third and slightly more complex expression: $2 \cdot 2^2 \cdot 2^{2^2}:\left[2+2^2+2^{2^2}:\left(2^{2^2}-2^2 \cdot 2\right)\right]$. This expression includes exponents and multiple levels of parentheses and brackets, so careful calculation is essential. Remember to work from the innermost parentheses outwards and to follow the order of operations. This problem demonstrates the importance of detailed, systematic approach to solving complex mathematical problems.

Step-by-Step Solution

1. Simplify the innermost parentheses: Let's look at the innermost part, $\left(2^{2^2}-2^2 \cdot 2\right)$. We calculate $2^2 = 4$, so the expression becomes $(2^4 - 4 \cdot 2)$. Next, $2^4 = 16$ and $4 \cdot 2 = 8$, so we get $(16 - 8) = 8$.
2. Address the next level of parentheses: The expression now looks like this: $2 \cdot 2^2 \cdot 2^{2^2}:\left[2+2^2+2^{2^2}:8\right]$. Calculate $2^{2^2}$, which is $2^4 = 16$. The expression becomes $2 \cdot 2^2 \cdot 16:\left[2+4+16:8\right]$.
3. Simplify within the brackets: Now, let's address what's inside the square brackets. We have $2+4+16:8$. First, $16:8 = 2$, so we have $2 + 4 + 2 = 8$.
4. Simplify the numerator: Next, consider $2 \cdot 2^2 \cdot 16$. Calculate $2^2 = 4$, so we have $2 \cdot 4 \cdot 16 = 8 \cdot 16 = 128$.
5. Perform the final division: The expression now is $128:8$, which simplifies to 16.

Therefore, the final result of the entire expression is 16. Each step has been calculated individually and logically to make sure that the order of operations has been carefully observed. In each calculation, we have shown how carefully working through the order of operations can lead to the right answer. The correct answer has been achieved through careful calculation and step-by-step procedures. Congratulations! You've successfully navigated these complex math problems.