Calculating Expressions With Fractional Exponents

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Hey guys! Today, we're diving into the world of fractional exponents and tackling some calculations. Fractional exponents might seem a bit tricky at first, but once you understand the basics, you'll be solving these problems like a pro. We'll break down each expression step-by-step, so you can follow along easily. Let's jump right in!

Understanding Fractional Exponents

Before we get into the nitty-gritty calculations, let's quickly recap what fractional exponents actually mean. A fractional exponent like x^( m/n) represents both a power and a root. The numerator (m) indicates the power to which the base (x) is raised, and the denominator (n) indicates the root to be taken. So, x^( m/n) is the same as (n√(x))^m.

For example, if we have 8^(1/3), this means we're taking the cube root of 8. Since 2 * 2 * 2 = 8, the cube root of 8 is 2. So, 8^(1/3) = 2. Similarly, if we have (-32)^(1/5), we're looking for a number that, when multiplied by itself five times, equals -32. In this case, it's -2 because (-2) * (-2) * (-2) * (-2) * (-2) = -32.

Understanding this concept is crucial for solving the expressions we have at hand. Remember, when dealing with negative bases and fractional exponents, especially when the denominator is an even number, we need to be careful about whether the result will be a real number. However, in our examples today, we'll primarily be dealing with cases where the roots are real numbers.

Solving the Expressions

Now, let's get to the main part – calculating the expressions. We'll go through each one step by step, so you can see exactly how we arrive at the solution. Make sure you have your pen and paper ready, and let's start!

a) 1/2 * (-1)^(1/3) + 1/3 * (-32)^(1/5)

In this expression, we need to calculate two parts involving fractional exponents and then add them together. Let's break it down:

  • (-1)^(1/3): This means we need to find the cube root of -1. What number, when multiplied by itself three times, equals -1? The answer is -1 because (-1) * (-1) * (-1) = -1.
  • (-32)^(1/5): Here, we need to find the fifth root of -32. This means we're looking for a number that, when multiplied by itself five times, gives us -32. As we discussed earlier, this number is -2.

Now, let's substitute these values back into the expression:

1/2 * (-1) + 1/3 * (-2)

Next, we perform the multiplications:

  • 1/2 * (-1) = -1/2
  • 1/3 * (-2) = -2/3

So, our expression now looks like this:

-1/2 + (-2/3)

To add these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6. So, we convert the fractions:

  • -1/2 = -3/6
  • -2/3 = -4/6

Now we can add them:

-3/6 + (-4/6) = -7/6

So, the final answer for part a) is -7/6, which can also be written as -1 1/6.

b) 3 * (-0.125)^(1/3) - 2 * (-125)^(1/3)

For this expression, we again have to deal with cube roots, but this time we have decimals and larger numbers. Let's tackle it:

  • (-0.125)^(1/3): We need to find the cube root of -0.125. It's helpful to recognize that 0.125 is 1/8. So, we're looking for the cube root of -1/8. Since (-1/2) * (-1/2) * (-1/2) = -1/8, the cube root of -0.125 is -0.5.
  • (-125)^(1/3): This is the cube root of -125. We need a number that, when multiplied by itself three times, equals -125. That number is -5, because (-5) * (-5) * (-5) = -125.

Now, let's plug these values back into the expression:

3 * (-0.5) - 2 * (-5)

Perform the multiplications:

  • 3 * (-0.5) = -1.5
  • 2 * (-5) = -10

So, the expression becomes:

-1.5 - (-10)

Subtracting a negative is the same as adding, so:

-1.5 + 10 = 8.5

Therefore, the final answer for part b) is 8.5.

c) 3 * (-0.008)^(1/3) * 5 * (-0.00032)^(1/5)

This expression involves both a cube root and a fifth root. Let's break it down:

  • (-0.008)^(1/3): We need the cube root of -0.008. Recognizing that 0.008 is 8/1000, we can rewrite it as -8/1000. The cube root of -8 is -2, and the cube root of 1000 is 10. So, the cube root of -0.008 is -2/10, which simplifies to -0.2.
  • (-0.00032)^(1/5): We need the fifth root of -0.00032. This might look tricky, but let's think of 0.00032 as 32/100000, which simplifies to 1/3125. So we are looking for the fifth root of -1/3125. Since (-1/5) * (-1/5) * (-1/5) * (-1/5) * (-1/5) = -1/3125, the fifth root of -0.00032 is -0.2.

Now, substitute these values back into the expression:

3 * (-0.2) * 5 * (-0.2)

Perform the multiplications:

  • 3 * (-0.2) = -0.6
  • 5 * (-0.2) = -1

So, the expression becomes:

-0.6 * (-1)

Multiply:

-0.6 * (-1) = 0.6

Thus, the final answer for part c) is 0.6.

d) (-2)^(1/3) * (-500)^(1/3)

Here, we have the product of two cube roots. We can use the property that a^(1/n) * b^(1/n) = (a * b)^(1/n). So:

(-2)^(1/3) * (-500)^(1/3) = (-2 * -500)^(1/3)

Multiply -2 and -500:

-2 * -500 = 1000

Now we have:

(1000)^(1/3)

We need the cube root of 1000. What number, when multiplied by itself three times, equals 1000? It's 10, because 10 * 10 * 10 = 1000.

So, the final answer for part d) is 10.

e) (-56)^(1/3) : (7)^(1/3)

For this expression, we're dividing two cube roots. Similar to multiplication, we can use the property that a^(1/n) / b^(1/n) = (a/b)^(1/n). So:

(-56)^(1/3) / (7)^(1/3) = (-56 / 7)^(1/3)

Divide -56 by 7:

-56 / 7 = -8

Now we have:

(-8)^(1/3)

We need the cube root of -8. What number, when multiplied by itself three times, equals -8? It's -2, because (-2) * (-2) * (-2) = -8.

So, the final answer for part e) is -2.

Conclusion

And there you have it! We've successfully calculated all the expressions involving fractional exponents. Remember, the key is to understand what fractional exponents mean and to break down each problem into smaller, manageable steps. With practice, you'll become more confident in handling these types of calculations. Keep up the great work, guys, and happy calculating!