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Chapter 1: Problem 31

Simplify the given expressions. In each of \(5-9\) and \(12-21,\) the result is an integer. $$\sqrt[3]{-8^{2}}$$

Short Answer

Expert verified
The expression simplifies to 4.

Step by step solution

01

Understand the Expression

We need to simplify the expression \( \sqrt[3]{-8^2} \). It involves finding the cube root of a negative number raised to a power.
02

Calculate 8^2

First, calculate \(-8^2\). Squaring \(-8\) gives us:\[-8^2 = (-8) \times (-8) = 64.\]
03

Simplify the Expression

Now, the expression becomes \( \sqrt[3]{64} \), as we calculated \(-8^2\) to be \(64\).
04

Calculate the Cube Root

Next, find the cube root of \(64\). We know that:\[4 \times 4 \times 4 = 64\]Thus, the cube root of \(64\) is \(4\).

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Key Concepts

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

Understanding Cube Roots
Cube roots are all about finding a number that, when multiplied by itself twice, results in the original number given. For instance, if you have a number like 64 and you want to find its cube root, you're essentially asking: "What number multiplied by itself three times equals 64?" If the number you're looking for is 4, then it must be because the calculation is:
  • 4 multiplied by 4 gives 16.
  • Multiply 16 by 4 again, and you get 64.
Thus, the cube root of 64 is 4. Cube roots can be thought of as the opposite of cubing (raising a number to the power of three). When you see the term cube root, think about finding a smaller, repetitive factor of the larger number.
Achieving an Integer Result
An integer result means your answer, after all calculations, is a whole number (positive, negative, or zero), without any fractions or decimals. In problems involving cube roots, we're often asked to simplify to an integer.
For example, when simplifying \[\sqrt[3]{64}\]we seek an integer value for which \[x^3 = 64.\]After working through the math, we know this integer must be 4, since \[4^3 = 64.\]When simplifying such expressions, it's helpful to recognize cube numbers or memorize at least the first few cube results, like:
  • 1 cubed is 1
  • 2 cubed is 8
  • 3 cubed is 27
  • 4 cubed is 64
  • 5 cubed is 125
This knowledge can speed up the process of finding the integer result.
Handling Negative Numbers
Negative numbers can be tricky, but understanding how they work with operations like squaring and cubing is vital. When you square a negative number, like \(-8\), you're multiplying it by itself:
  • \((-8) \times (-8) = 64.\)
The negative signs cancel each other out, resulting in a positive number. However, if it were cubed, the result would revert to negative:
  • \(-8 \times -8 \times -8 = -512.\)
This is because multiplying an odd number of negative factors yields a negative product. Understanding this helps with simplifying expressions as it guides correct sign usage.
The Role of Exponentiation
Exponentiation involves repeatedly multiplying a number by itself. In the expression \(-8^2\), exponentiation tells us to multiply \(-8\) by itself:
  • \(-8 \times -8 = 64.\)
Here, the exponent (2) acts as a directive on how many times to use the number \(-8\) in multiplication. The power of exponents is that they allow us to express repeated multiplication concisely.
With negative numbers, whether the result is positive or negative depends on whether the exponent is even or odd. Exponentiation is a fundamental process in math that simplifies expressions and reveals patterns in number operations.

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Most popular questions from this chapter

For any real numbers \(a\) and \(b\), is it true that \(|a-b|=|b-a| ?\) Explain.

Perform the indicated multiplications. The weekly revenue \(R\) (in dollars) of a flash drive manufacturer is given by \(R=x p,\) where \(x\) is the number of flash drives sold each week and \(p\) is the price (in dollars). If the price is given by the demand equation \(p=30-0.01 x,\) express the revenue in terms of \(x\) and simplify.

Perform the indicated operations assuming all numbers are approximate. Round your answers using the procedure shown in Example 11 $$4.52-\frac{2.056(309.6)}{395.2}$$

To find the amount of a certain investment of \(x\) dollars, it is necessary to solve the equation \(0.03 x+0.06(2000-x)=96 .\) Solve for \(x.\)

To find the voltage \(V\) in a circuit in a TV remote-control unit, the equation \(1.12 V-0.67(10.5-V)=0\) is used. Find \(V.\)
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