Calculate 1.6^7 * 0.625^13 / (-3-1.4)^3 A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun algebra problem. We need to calculate the value of the expression: 1.6^7 * 0.625^13 / (-3 - 1.4)^3. Don't worry, it looks intimidating, but we'll break it down step by step. Let's make it super easy to understand. We will go through the properties of exponents and fractions to see how we can simplify this expression and arrive at the answer. This calculation requires us to understand how to work with exponents, decimals, and negative numbers. So, let's get started and make math a little less scary, alright?

Step-by-Step Breakdown

First, let's rewrite the numbers in fraction form and simplify. Rewriting decimals as fractions often makes the calculations easier. So, we can express 1.6 as 16/10, which simplifies to 8/5. Similarly, 0.625 can be written as 625/1000, which simplifies to 5/8. Next, we'll simplify the expression inside the parenthesis in the denominator. Remember guys, working with fractions and exponents is like building blocks—each step gets us closer to the final answer. We'll also explore some properties of exponents to see how they apply here. Now, the expression looks like this:

(8/5)^7 * (5/8)^13 / (-3 - 1.4)^3

Now, let's focus on the denominator. We have (-3 - 1.4)^3. We need to simplify the expression inside the parentheses first. Combining -3 and -1.4, we get -4.4. So, now we have (-4.4)^3. To make things easier, let's convert -4.4 into a fraction. -4.4 is the same as -44/10, which simplifies to -22/5. So, the expression in the denominator becomes (-22/5)^3. Guys, are you following along? It's like we're detectives solving a mathematical mystery!

Now, our expression looks like this:

(8/5)^7 * (5/8)^13 / (-22/5)^3

Next, we'll use the properties of exponents to simplify the numerator. We have (8/5)^7 and (5/8)^13. Notice that 5/8 is the reciprocal of 8/5. This is a crucial observation because it allows us to simplify the expression further. We can rewrite (5/8)^13 as (8/5)^-13. Do you see where we are going with this? When you have exponents, think of them as instructions that tell you how many times to multiply the base by itself. Let's keep going!

So, now the expression becomes:

(8/5)^7 * (8/5)^-13 / (-22/5)^3

When you multiply numbers with the same base, you add the exponents. So, (8/5)^7 * (8/5)^-13 becomes (8/5)^(7 + (-13)), which simplifies to (8/5)^-6. Guys, we're making real progress here! It's like we're untangling a mathematical knot, one step at a time. Always remember, exponents can be added when you're multiplying powers with the same base.

Now, our expression looks like this:

(8/5)^-6 / (-22/5)^3

Now, let's handle the negative exponent. A negative exponent means that we take the reciprocal of the base and raise it to the positive exponent. So, (8/5)^-6 becomes (5/8)^6. Remember, guys, negative exponents can be a bit tricky, but they're just another tool in our math kit. They tell us to flip the fraction and then apply the positive exponent.

Now, our expression looks like this:

(5/8)^6 / (-22/5)^3

Okay, now we're ready to deal with the denominator. We have (-22/5)^3. This means we need to multiply -22/5 by itself three times. When you raise a negative number to an odd power, the result will be negative. So, (-22/5)^3 is equal to -(22^3 / 5^3). Calculating 22^3 gives us 10648, and 5^3 is 125. So, (-22/5)^3 is -10648/125. You're doing great! Remember to take your time and double-check your calculations. It's like building a puzzle, each piece needs to fit just right.

Now, our expression looks like this:

(5/8)^6 / (-10648/125)

To simplify (5/8)^6, we raise both the numerator and the denominator to the power of 6. So, we have (5^6 / 8^6). Calculating these values, we get 5^6 = 15625 and 8^6 = 262144. So, (5/8)^6 is 15625/262144. Guys, we're almost there! We've tackled the exponents, and now we're ready to divide fractions.

Now, our expression looks like this:

(15625/262144) / (-10648/125)

Dividing by a fraction is the same as multiplying by its reciprocal. So, we need to multiply 15625/262144 by the reciprocal of -10648/125, which is -125/10648. Remember guys, dividing fractions is just multiplying by the inverse. It's like a mathematical dance, swapping the numerator and denominator to get the job done.

So, the expression becomes:

(15625/262144) * (-125/10648)

Multiplying the numerators and the denominators, we get (15625 * -125) / (262144 * 10648). This simplifies to -1953125 / 2791478272. Guys, we're on the home stretch! Just a bit more simplification, and we'll have our final answer.

Final Simplification and Answer

Now, we have the fraction -1953125 / 2791478272. To simplify this fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. This can be a bit tricky, but we can use prime factorization to help us. After some calculations, we find that both numbers are divisible by 125. Dividing both the numerator and the denominator by 125, we get -15625 / 22331826.

So, the final simplified answer is approximately -0.000699. Guys, give yourselves a pat on the back! We made it through this complex calculation together. Remember, math is like a muscle, the more we exercise it, the stronger it gets. Keep practicing, and you'll be solving even tougher problems in no time!

Conclusion

In this exercise, we walked through how to simplify a complex expression involving exponents, fractions, and negative numbers. Remember, the key is to break the problem down into manageable steps. We started by rewriting decimals as fractions, simplified the exponents, and then performed the division. Always take your time, double-check your calculations, and don't be afraid to make mistakes—that's how we learn. And hey, guys, math can actually be fun, especially when you tackle a tough problem and come out on top. Keep exploring, keep learning, and keep those mathematical muscles strong!