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Chapter 5: Problem 3

Write each expression as a cube. $$-27$$

Short Answer

Expert verified
-27 can be expressed as \((-3)^3\).

Step by step solution

01

Find Cube root

In this step, find the cube root of the given number, -27. Cube root of a number can be found by determining which number, when multiplied by itself twice, equals the original number. In this case, the cube root of -27 is -3 because (-3)*(-3)*(-3)=-27.
02

Written as cube

In the previous step, it was found that the cube root of -27 is -3. Therefore, -27 can be written as the cube of -3. In mathematical notation, this is expressed as \((-3)^3\).

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Key Concepts

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

Cube Notation
In mathematics, expressing a number as a cube involves using cube notation. Cube notation is a form of expressing a number that is the result of multiplying a number by itself two more times.
Basically, when you see a number as, for example, \(a^3\), this means \(a \times a \times a\).
  • In our example, the number -27 is expressed as the cube of -3.
  • Writing it in cube notation becomes \((-3)^3\).
The idea is to understand that a cube is simply a number multiplied by itself twice, and cube notation makes it easier to convey this idea in a concise way.
Negative Numbers
Handling negative numbers can be tricky, especially when dealing with operations such as cubes.
Negative numbers have been a part of mathematics for a long time, and they behave in specific ways during operations.
  • When multiplying negative numbers, we must remember that a negative number multiplied by a negative number will yield a positive result.
  • However, if we multiply a negative number by itself an odd number of times, such as in cubing, the result remains negative.
In our exercise, \(-3 \times -3 \times -3\) gives us -27 because two negatives make a positive, but the third multiplication with a negative turns the result back to negative again.
Exponents
An exponent indicates how many times a number, known as the base, is multiplied by itself.
It is a crucial mathematical concept for operations including squares and cubes.
  • Here, the cube is an exponent of 3, meaning the number is used three times as a factor.
  • For anyone dealing with exponents, especially with negative numbers, it's helpful to remember the properties of exponents.
In case of \((-3)^3\), the exponent 3 dictates that -3 is multiplied by itself two more times. Thus, understanding exponents helps clarify how expressions like \((-3)^3\) compute to -27.

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Most popular questions from this chapter

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$-3 y^{3}+6 y^{2}+3 y$$

Factor. $$16 x^{2}+8 x y+y^{2}-9$$

When we add \(5 x\) and \(7 x\), we combine like terms: \(5 x+7 x=12 x\). Explain how this is related to factoring out a common factor.

Factor each expression completely. $$2 r-b s-2 s+b r$$

Factor. $$9 x^{2}-6 x+1-d^{2}$$
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