Solving Exponential Math Expressions: Step-by-Step Guide

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Hey guys! Today, we're diving into some exciting math problems involving exponents. Don't worry if these look intimidating at first – we'll break them down step by step to make them super easy to understand. We'll be tackling four expressions (a, b, c, and d) that involve different operations with exponents. So, grab your calculators (or your mental math skills!), and let’s get started!

a) 3^3 - 3^2 : (2^6 - 2^5 - 2^4 - 2^3 - 2^2 - 2^1 - 2^0)

Let's kick things off with our first expression. This one looks a bit complex, but we can solve it by following the order of operations (PEMDAS/BODMAS). Remember, this means we handle parentheses/brackets first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

Breaking Down the Expression

Our main goal here is to simplify the expression piece by piece. We'll start by calculating each exponential term individually and then work our way through the parentheses and the rest of the equation.

  1. Calculate the Powers of 3:

    • 3^3 means 3 multiplied by itself three times, which is 3 * 3 * 3 = 27. So, 3^3 = 27.
    • 3^2 means 3 multiplied by itself twice, which is 3 * 3 = 9. Thus, 3^2 = 9.

  2. Calculate the Powers of 2 inside the Parentheses:

    • 2^6 means 2 multiplied by itself six times, which is 2 * 2 * 2 * 2 * 2 * 2 = 64. So, 2^6 = 64.
    • 2^5 means 2 multiplied by itself five times, which is 2 * 2 * 2 * 2 * 2 = 32. Thus, 2^5 = 32.
    • 2^4 means 2 multiplied by itself four times, which is 2 * 2 * 2 * 2 = 16. Hence, 2^4 = 16.
    • 2^3 means 2 multiplied by itself three times, which is 2 * 2 * 2 = 8. So, 2^3 = 8.
    • 2^2 means 2 multiplied by itself twice, which is 2 * 2 = 4. Therefore, 2^2 = 4.
    • 2^1 means 2 to the power of 1, which is just 2. So, 2^1 = 2.
    • 2^0 means 2 to the power of 0, and anything to the power of 0 is 1. Thus, 2^0 = 1.

  3. Substitute the Values into the Expression: Now that we've calculated all the powers, let's plug them back into the original expression: 27 - 9 : (64 - 32 - 16 - 8 - 4 - 2 - 1)

  4. Simplify Inside the Parentheses: Next, we'll simplify the expression inside the parentheses: 64 - 32 - 16 - 8 - 4 - 2 - 1 = 1

  5. Perform the Division: Now we have: 27 - 9 : 1 Division comes before subtraction, so we'll divide 9 by 1: 9 : 1 = 9

  6. Perform the Subtraction: Finally, subtract 9 from 27: 27 - 9 = 18

Final Result

So, the result of the expression 3^3 - 3^2 : (2^6 - 2^5 - 2^4 - 2^3 - 2^2 - 2^1 - 2^0) is 18. Yay, we did it!

b) (5^7 + 5^6 + 5^5) : 5^5 + 9^7 : 3^{10}

Moving on to our second expression, we're dealing with a combination of powers and division. This one might seem tricky, but we'll use some factoring and exponent rules to simplify it effectively.

Simplifying the Expression

Our strategy here involves factoring out common terms and using exponent rules to reduce the expression to a manageable form. We'll also handle the two main parts of the expression separately before combining them.

  1. Factor out 5^5 from the First Part: The first part of the expression is (5^7 + 5^6 + 5^5) : 5^5. We can factor out 5^5 from the terms inside the parentheses: (5^7 + 5^6 + 5^5) = 5^5 * (5^2 + 5^1 + 1) Now, let's substitute this back into the first part of the expression: (5^5 * (5^2 + 5^1 + 1)) : 5^5

  2. Simplify the First Part: We can now divide the entire expression by 5^5: (5^5 * (5^2 + 5^1 + 1)) : 5^5 = 5^2 + 5^1 + 1 Calculate the powers: 5^2 = 25 5^1 = 5 So, the simplified form is: 25 + 5 + 1 = 31

  3. Simplify the Second Part: The second part of the expression is 9^7 : 3^10}. Notice that 9 is 3^2, so we can rewrite 9^7 as (32)7. Using the power of a power rule (a{m}){n} = a^{mn}, we get *(32)7 = 3^{14 Now, the second part of the expression becomes: 3^14} 3^{10 Using the quotient of powers rule (a^m : a^n = a^m-n})*, we get *3^{14 : 3^{10} = 3^{14-10} = 3^4 Calculate 3^4: 3^4 = 3 * 3 * 3 * 3 = 81

  4. Combine the Simplified Parts: Now we add the results from the simplified first and second parts: 31 + 81 = 112

Final Result

So, the final result of the expression (5^7 + 5^6 + 5^5) : 5^5 + 9^7 : 3^{10} is 112. Awesome!

c) 2 * 2^2 * 2{22} : [2 + 2^2 + 2{22} : (2{22} - 2^2 * 2)]

Our third expression is a bit of a beast, but don't sweat it! It involves nested exponents and parentheses, so we'll need to be extra careful with the order of operations. Let's break it down systematically.

Step-by-Step Simplification

We'll start by simplifying the exponents and working our way from the innermost parentheses outwards. Patience is key here!

  1. Calculate the Innermost Exponents: First, we need to evaluate 2^2: 2^2 = 2 * 2 = 4 Now, substitute this back into the expression: 2 * 2^2 * 2^4 : [2 + 2^2 + 2^4 : (2^4 - 2^2 * 2)]

  2. Calculate 2^4: 2^4 = 2 * 2 * 2 * 2 = 16 Substitute 2^4 back into the expression: 2 * 2^2 * 16 : [2 + 2^2 + 16 : (16 - 2^2 * 2)]

  3. Calculate 2^2 in the Parentheses: 2^2 = 4 Substitute it back: 2 * 2^2 * 16 : [2 + 4 + 16 : (16 - 4 * 2)]

  4. Simplify Inside the Innermost Parentheses: First, perform the multiplication: 4 * 2 = 8 Substitute it back: 2 * 2^2 * 16 : [2 + 4 + 16 : (16 - 8)] Now, perform the subtraction: 16 - 8 = 8 Substitute it back: 2 * 2^2 * 16 : [2 + 4 + 16 : 8]

  5. Simplify Inside the Brackets: First, perform the division: 16 : 8 = 2 Substitute it back: 2 * 2^2 * 16 : [2 + 4 + 2] Now, add the numbers inside the brackets: 2 + 4 + 2 = 8 Substitute it back: 2 * 2^2 * 16 : 8

  6. Calculate 2^2: 2^2 = 4 Substitute it back: 2 * 4 * 16 : 8

  7. Perform Multiplication from Left to Right: First multiplication: 2 * 4 = 8 Substitute it back: 8 * 16 : 8 Second multiplication: 8 * 16 = 128 Substitute it back: 128 : 8

  8. Perform the Division: 128 : 8 = 16

Final Result

So, the final result of the expression 2 * 2^2 * 2{22} : [2 + 2^2 + 2{22} : (2{22} - 2^2 * 2)] is 16. Whew, that was a workout!

d) 3^2 + (4 - 6)^2 + 2^5 * 5^2 - 1

Last but not least, we have our fourth expression. This one involves a mix of exponents, subtraction, multiplication, and addition. We'll tackle it using our trusty order of operations.

Step-by-Step Solution

We'll follow the PEMDAS/BODMAS rule to ensure we simplify the expression correctly.

  1. Calculate Inside the Parentheses: First, handle the expression inside the parentheses: 4 - 6 = -2 So, the expression becomes: 3^2 + (-2)^2 + 2^5 * 5^2 - 1

  2. Calculate the Exponents: 3^2 = 3 * 3 = 9 (-2)^2 = -2 * -2 = 4 2^5 = 2 * 2 * 2 * 2 * 2 = 32 5^2 = 5 * 5 = 25 Substitute the values back into the expression: 9 + 4 + 32 * 25 - 1

  3. Perform the Multiplication: Multiply 32 by 25: 32 * 25 = 800 Substitute it back: 9 + 4 + 800 - 1

  4. Perform Addition and Subtraction from Left to Right: First addition: 9 + 4 = 13 Substitute it back: 13 + 800 - 1 Second addition: 13 + 800 = 813 Substitute it back: 813 - 1 Final subtraction: 813 - 1 = 812

Final Result

So, the final result of the expression 3^2 + (4 - 6)^2 + 2^5 * 5^2 - 1 is 812. Fantastic job!

Conclusion

And there you have it! We've successfully solved four complex expressions involving exponents. Remember, the key to mastering these types of problems is to break them down step by step, follow the order of operations, and utilize exponent rules whenever possible. Keep practicing, and you'll become an exponent whiz in no time. You got this!