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Chapter 7: Problem 32

Evaluate expression. \(1000^{2 / 3}\)

Short Answer

Expert verified
100

Step by step solution

01

Understand the Exponent

The exponent \(\frac{2}{3}\) indicates that we need to take the cube root of 1000 and then square the result.
02

Take the Cube Root of 1000

We start by finding the cube root of 1000. We need to determine which number, when raised to the power of 3, gives us 1000: \(\root3\relax{1000}\). Since \(\root3\relax{1000} = 10\), we have that 10 is the cube root of 1000.
03

Square the Result

Now we take the result from Step 2 and square it: \((10)^{2}\). Squaring 10 gives us 100.

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Key Concepts

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

cube root
First, we'll cover the concept of a cube root. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. In mathematical terms, if \(\root3\relax{x} = y\), then \(y^3 = x\).
For example:
  • The cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\)
  • The cube root of 64 is 4, because \(4 \times 4 \times 4 = 64\)

When dealing with fractional exponents, such as \(1000^{\frac{2}{3}}\), the denominator (3) tells us to first find the cube root of 1000.
squaring
Next, let's talk about squaring numbers. Squaring a number means multiplying it by itself. In other words, if \(x\) is our number, squaring it means performing \(x \times x\) or \(x^2\).
For example:
  • \(2^2 = 4\)
  • \(5^2 = 25\)

Returning to our problem with the cube root already found, we had \(10\) from the cube root of 1000. Next, we need to square 10. This is done simply by performing \(10^2 = 100\).
evaluating expressions
Finally, let's understand how to evaluate expressions, particularly those with fractional exponents. Evaluating an expression means finding its value. When we have a fractional exponent like \(1000^{\frac{2}{3}}\), we need to break it down.
Start by:
  • Identifying the Cube Root
  • Then Squaring the Result

In this example, the fraction \( \frac{2}{3} \) tells us to take the cube root of 1000 first, which is 10, and then square that result to get 100.
Step-by-step, it goes as follows:
  • Find \(\root3\relax{1000} = 10\)
  • Then square 10: \(10^2 = 100\)
This is how we simplify and solve \(1000^{\frac{2}{3}} = 100\).

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Most popular questions from this chapter

Solve each equation. $$2 x^{2}+3=7$$

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Simplify. $$\frac{3}{\sqrt{2}-1}+\frac{4}{\sqrt{2}+1}$$
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