Calculating Exponential Expressions: Step-by-Step Solutions

Oct 29, 2025 by SLV Team 60 views

Hey guys! Today, we're diving into the exciting world of exponential expressions. We'll break down how to calculate several expressions involving exponents, making sure you understand each step along the way. Whether you're a student tackling homework or just brushing up on your math skills, this guide is for you. So, let's jump right in and get those exponents figured out!

a) (2^9 - 3^7) : (2^8 - 3^7)

When we first look at this expression, (2^9 - 3^7) : (2^8 - 3^7), it might seem a little intimidating. But don't worry, we'll take it one step at a time. The key here is to calculate the values of the exponents first.

First, let's calculate 2^9, which means 2 multiplied by itself 9 times: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2. If you do the math, you'll find that 2^9 = 512.
Next, we need to calculate 3^7, which means 3 multiplied by itself 7 times: 3 * 3 * 3 * 3 * 3 * 3 * 3. This gives us 3^7 = 2187.
Now, we can substitute these values back into our expression. So, (2^9 - 3^7) becomes (512 - 2187).

Let's do the subtraction: 512 - 2187 = -1675. So, the numerator of our expression is -1675.

Now, let's move on to the denominator, which is (2^8 - 3^7). We already know that 3^7 = 2187, so we just need to calculate 2^8.

2^8 means 2 multiplied by itself 8 times: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2. This equals 256.
Substituting this value, (2^8 - 3^7) becomes (256 - 2187).

Subtracting these numbers, we get 256 - 2187 = -1931. So, the denominator is -1931.

Now, our original expression (2^9 - 3^7) : (2^8 - 3^7) has been simplified to (-1675) : (-1931). To get the final answer, we divide -1675 by -1931.

Dividing -1675 by -1931 gives us approximately 0.867. Since both numbers are negative, the result is positive.

So, the final answer for part a) is approximately 0.867. Remember, breaking down the problem into smaller steps makes it much easier to solve!

b) (2^12 - 3^41) : (2^10 - 3^40)

Okay, let's tackle the second expression, (2^12 - 3^41) : (2^10 - 3^40). This one looks a bit more complex due to the larger exponents, but we'll handle it step by step.

First off, calculating 3^41 and 3^40 directly would give us extremely large numbers, likely requiring advanced calculators or software. Similarly, 2^12 isn't too bad, but 3^41 is huge! Because of the impracticality of manual computation in this case, we can recognize that without computational tools, expressing the result in its exact numerical form is challenging. We'll note that for a complete solution, one would typically use computational software or a high-powered calculator.

However, let's focus on breaking down what we can. 2^12 is manageable:

2^12 means 2 multiplied by itself 12 times: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2. This equals 4096.
Similarly, 2^10 means 2 multiplied by itself 10 times, which equals 1024.

So, we have 2^12 = 4096 and 2^10 = 1024. We can substitute these values back into our expression, but we'll still have to deal with 3^41 and 3^40.

Our expression now looks like this: (4096 - 3^41) : (1024 - 3^40).

Without the means to compute 3^41 and 3^40, we can express the answer symbolically, acknowledging that a numerical solution would require computational assistance. So, the practical approach here is to leave it in the form we have after substituting the feasible computations.

Thus, for a numerical result, computational tools are essential, but understanding the breakdown is key. The final computed answer would be a decimal close to 0 due to the sheer magnitude of 3^41 and 3^40 dominating the equation.

c) (2^11 * 3^2 * 5^7) : (2^10 * 3^2 * 5^7)

Alright, let's move on to the third expression: (2^11 * 3^2 * 5^7) : (2^10 * 3^2 * 5^7). This one might look complicated at first, but it's actually a great example of how simplifying expressions can be easier than you think.

When we're dividing terms with exponents, we can use some handy rules to simplify things. In this case, we have several common factors in both the numerator and the denominator. Let's break it down:

We have 2^11 in the numerator and 2^10 in the denominator. Remember the rule for dividing exponents with the same base: a^m / a^n = a^(m-n). So, 2^11 / 2^10 = 2^(11-10) = 2^1 = 2.
Next, we have 3^2 in both the numerator and the denominator. When we divide something by itself, the result is 1. So, 3^2 / 3^2 = 1.
Similarly, we have 5^7 in both the numerator and the denominator. Again, when we divide something by itself, the result is 1. So, 5^7 / 5^7 = 1.

Now, let's put it all together. Our original expression (2^11 * 3^2 * 5^7) : (2^10 * 3^2 * 5^7) simplifies to:

2 * 1 * 1 = 2

So, the final answer for part c) is simply 2. See? Sometimes these expressions are hiding a simple solution!

d) (2^9 * 3^3 * 5^6) : (2^8 * 5^6)

Let's dive into the fourth expression: (2^9 * 3^3 * 5^6) : (2^8 * 5^6). This one is similar to the previous example, where we can simplify by using the rules of exponents. It involves dividing terms with the same base, so let's break it down piece by piece.

First, let's look at the terms with the base 2: we have 2^9 in the numerator and 2^8 in the denominator. As we saw before, when dividing exponents with the same base, we subtract the exponents: a^m / a^n = a^(m-n). So, 2^9 / 2^8 = 2^(9-8) = 2^1 = 2.
Next, we have 3^3 in the numerator. There's no term with the base 3 in the denominator, so we'll just leave it as is for now.
Now, let's look at the terms with the base 5: we have 5^6 in both the numerator and the denominator. Just like before, when we divide something by itself, the result is 1. So, 5^6 / 5^6 = 1.

Now, let's put it all together. Our original expression (2^9 * 3^3 * 5^6) : (2^8 * 5^6) simplifies to:

2 * 3^3 * 1

We still need to calculate 3^3, which means 3 multiplied by itself 3 times: 3 * 3 * 3 = 27.

So, our expression becomes:

2 * 27 * 1

Multiplying these numbers, we get:

2 * 27 = 54

So, the final answer for part d) is 54. By simplifying each part of the expression, we made it much easier to solve.

e) (2^2 - 3)^12 : (2^24 * 3^12)

Now, let's tackle expression e): (2^2 - 3)^12 : (2^24 * 3^12). This one looks a bit tricky, but we'll break it down step by step to make it manageable.

First, let's focus on the term inside the parentheses: (2^2 - 3). We need to calculate 2^2, which means 2 multiplied by itself: 2 * 2 = 4. So, 2^2 = 4.

Now we can substitute this value back into the parentheses: (4 - 3) = 1. So, (2^2 - 3) simplifies to 1.

Now our expression looks like this: 1^12 : (2^24 * 3^12). We know that any number raised to the power of 12 remains the same, so 1^12 = 1.

Our expression now simplifies to: 1 : (2^24 * 3^12)

Next, let's look at the denominator: 2^24 * 3^12. We can rewrite 2^24 as (2²⁾12, which is 4^12. So, our denominator becomes 4^12 * 3^12.

Now we can rewrite our expression as:

1 / (4^12 * 3^12)

Using the property of exponents, a^n * b^n = (a * b)^n, we can combine 4^12 and 3^12:

4^12 * 3^12 = (4 * 3)^12 = 12^12

So, our expression simplifies to:

1 / 12^12

This is a very small fraction, but it's our final answer. In decimal form, this would be an extremely small number. We can leave the answer as 1 / 12^12 or calculate the decimal value if needed.

So, the final answer for part e) is 1 / 12^12. By breaking down the expression and simplifying each part, we were able to solve it step by step.

f) (2^31 - 3^52) : (2^8 - 3⁵⁾10

Lastly, let's tackle expression f): (2^31 - 3^52) : (2^8 - 3⁵⁾10. This one looks quite intimidating, with large exponents and multiple operations. But don't worry, we'll take it one step at a time.

First, let's focus on the term inside the parentheses in the denominator: (2^8 - 3^5). We need to calculate 2^8 and 3^5.

2^8 means 2 multiplied by itself 8 times: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 256. So, 2^8 = 256.
3^5 means 3 multiplied by itself 5 times: 3 * 3 * 3 * 3 * 3 = 243. So, 3^5 = 243.

Now we can substitute these values back into the parentheses: (256 - 243) = 13. So, (2^8 - 3^5) simplifies to 13.

Our expression now looks like this: (2^31 - 3^52) : 13^10

Next, let's consider 2^31 and 3^52. These are very large numbers, and calculating them directly would be quite challenging without a calculator or computer. So, for practical purposes, we'll leave these terms as they are for now.

Our expression is now: (2^31 - 3^52) / 13^10

Without computational tools, we can't simplify this expression further to obtain a numerical value. The numerator involves subtracting two very large numbers, and the denominator is also a large number (13^10). However, we've simplified the expression as much as we can by hand.

For a complete numerical solution, you would typically use a calculator or computer software that can handle large numbers. But, understanding the breakdown and simplification steps is key.

So, the simplified expression for part f) is (2^31 - 3^52) / 13^10. Remember, sometimes the most important part is understanding how to break down the problem, even if we can't get a final numerical answer without additional tools.

Conclusion

And there you have it! We've walked through how to calculate several exponential expressions, breaking down each step to make it easier to understand. Remember, the key is to take it one step at a time, simplify where you can, and use the rules of exponents to your advantage. Whether it's simplifying complex expressions or finding simple solutions, practice makes perfect. Keep at it, and you'll become an exponent expert in no time!