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Chapter 7: Problem 99

Show that \(\left.\quad{ }^{3} \sqrt{(}-8\right)^{3}=\left({ }^{3} \sqrt{-} 8\right)^{3}\)

Short Answer

Expert verified
We are given the equation \[\left.\quad{ }^{3} \sqrt{-8}\right)^{3}=\left({ }^{3} \sqrt{-}8\right)^{3}\] and asked to show if it holds true. By finding the cube root of -8, which is -2, and then raising it to the power of 3, we get -8. Raising -8 to the power of 3 results in -512, and finding its cube root gives us -8. Since both results are equal (-8), the equation is true: \[\left.\quad{ }^{3} \sqrt{-8}\right)^{3}={ }^{3}\sqrt{(-8)^{3}}\].

Step by step solution

01

Find the cube root of -8

To find the cube root of -8, we can rewrite it as \((-8)^{\frac{1}{3}}\).
02

Evaluate (-8)^鈪�

Since we know that the cube root of any negative number is negative, and \(8^{\frac{1}{3}}=2\), we find that \({ }^{3}\sqrt{-8}=-2\).
03

Raise the cube root of -8 to the power of 3

Now, we need to calculate \[\left({ }^{3}\sqrt{-8}\right)^3\] which is \((-2)^3\).
04

Evaluate (-2)^3

-2 multiplied three times is -8, so \((-2)^3 = -8\).
05

Raise -8 to the power of 3

The next part of the equation is \[(-8)^3.\]
06

Evaluate (-8)^3

-8 multiplied three times is -512, so \((-8)^3 = -512\).
07

Find the cube root of -512

Now, we need to find the cube root of -512, as follows: \[{ }^{3}\sqrt{-512}\]
08

Evaluate \({ }^{3}\sqrt{-512}\)

Since the cube root of any negative number is negative, and \(512^{\frac{1}{3}}=8\), we find that \({ }^{3}\sqrt{-512}=-8\).
09

Compare the results

The results from steps 4 and 8 are equal as both are -8. Thus, the equation holds true: \[\left.\quad{ }^{3} \sqrt{-8}\right)^{3}={ }^{3}\sqrt{(-8)^{3}}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponents
Exponents are an essential concept in mathematics, allowing us to represent repeated multiplication concisely. When you see an expression like \((-8)^3\), it means that
  • -8 is multiplied by itself three times: \(-8 \times -8 \times -8\).
  • The result of this operation is -512.
Another way to understand exponents is by using cube roots. If you have a number raised to the power of \(1/3\), like \((-8)^{1/3}\), it indicates the cube root of that number. This is akin to finding what number, when multiplied by itself three times, gives the original number. For -8, that number is -2, as
  • \((-2) \times (-2) \times (-2) = -8\).
Understanding these operations is crucial, especially when dealing with more complex expressions.
Handling Negative Numbers
Negative numbers can sometimes be tricky, especially when combined with exponents. One key thing to remember is that the cube root of a negative number is also negative. This stems from the fact that multiplying three negative numbers together results in a negative number.

For example, consider the expression
  • \({ }^{3}\sqrt{-8}\).
This can be rewritten using exponents as \((-8)^{1/3}\) and simplifies to -2 because
  • \((-2) \times (-2) \times (-2) = -8\).
This contrasts with squaring negative numbers, where multiplying two negatives gives a positive result. Thus, it's essential to pay attention to whether the exponent is even or odd when working with negatives.
Evaluating Expressions with Exponents
Evaluating mathematical expressions requires careful attention to operations order, particularly with exponents. Start with simplifying inside the parentheses, then address exponents, such as cube roots or cubes, before proceeding with additional operations such as multiplication or addition.

In the context of cube roots and cubes:
  • First, evaluate the cube root, \((-8)^{1/3} = -2\).
  • Next, raise the result to the power of 3: \((-2)^3 = -8\).

Similarly, for \( (-8)^3 \), simply cube the number directly:
  • -8 multiplied three times gives -512.
By carefully following these steps, you can ensure accurate evaluation, maintaining clarity even in more complex problems.

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