Problem 30
Question
If possible, simplify the expression by hand. If you cannot, approximate the answer to the nearest hundredth. Variables represent any real number. $$ -\sqrt[3]{-1} $$
Step-by-Step Solution
Verified Answer
The simplified expression is 1.
1Step 1: Identify the Expression
The expression given is \(-\sqrt[3]{-1}\). This is the cube root of \(-1\) with a negative sign in front.
2Step 2: Calculate the Cube Root
The cube root of \(-1\) is the number that, when multiplied by itself three times, gives \(-1\). This number is \(-1\) because \((-1) \times (-1) \times (-1) = -1\).
3Step 3: Simplify the Expression
We apply the negative sign from the front of the cube root to the result of the cube root itself. Thus, the expression \(-\sqrt[3]{-1}\) simplifies to \(-(-1)\), which equals \(1\).
Key Concepts
Cube RootsReal NumbersNegative Numbers
Cube Roots
Understanding cube roots is key to tackling problems like the one presented. A cube root aims to find a number that, when multiplied by itself three times, returns the original number inside the root. For example, the cube root of 8 is 2, because \[2 \times 2 \times 2 = 8\].
One major difference between square roots and cube roots is how they handle negative numbers. While square roots of negative numbers are not real, cube roots can be real numbers even if the number inside is negative.
In the given exercise, finding the cube root of \(-1\) means looking for a number that satisfies: \[(-1) \times (-1) \times (-1) = -1\].
The answer is \(-1\), which may seem counterintuitive at first, but demonstrates the unique properties of cube roots.
One major difference between square roots and cube roots is how they handle negative numbers. While square roots of negative numbers are not real, cube roots can be real numbers even if the number inside is negative.
In the given exercise, finding the cube root of \(-1\) means looking for a number that satisfies: \[(-1) \times (-1) \times (-1) = -1\].
The answer is \(-1\), which may seem counterintuitive at first, but demonstrates the unique properties of cube roots.
Real Numbers
Real numbers are a broad category that includes most numbers we use in everyday life. They include rational numbers like fractions and whole numbers, as well as irrational numbers like \(\pi\) and the square root of 2.
Importantly, the concept of real numbers does not exclude negatives. This means that expressions involving cube roots, even of negative numbers, fall under the realm of real numbers. In the exercise, the expression simplifies to a real number, \(1\).
So, when solving problems involving the cube roots of negative numbers, remember that these situations still deal with real numbers. There's no need to venture into complex numbers, which are typically not real.
Importantly, the concept of real numbers does not exclude negatives. This means that expressions involving cube roots, even of negative numbers, fall under the realm of real numbers. In the exercise, the expression simplifies to a real number, \(1\).
So, when solving problems involving the cube roots of negative numbers, remember that these situations still deal with real numbers. There's no need to venture into complex numbers, which are typically not real.
Negative Numbers
Negative numbers play a significant role in mathematics, extending our understanding beyond zero. They are values less than zero, represented with a minus sign. In arithmetic operations, negative numbers follow unique rules, especially in multiplication and division.
For example, multiplying two negative numbers results in a positive number: \((-2) \times (-3) = 6\). However, multiplying an odd number of negative numbers keeps the product negative. So, \((-1) \times (-1) \times (-1) = -1\).
In the context of the original exercise, understanding how these rules apply to cube roots of negative numbers enables you to simplify expressions accurately. Recognizing that multiplying three negative numbers results in a negative number was key in calculating \(-\sqrt[3]{-1}\) and simplifying it to a positive result: \(1\).
For example, multiplying two negative numbers results in a positive number: \((-2) \times (-3) = 6\). However, multiplying an odd number of negative numbers keeps the product negative. So, \((-1) \times (-1) \times (-1) = -1\).
In the context of the original exercise, understanding how these rules apply to cube roots of negative numbers enables you to simplify expressions accurately. Recognizing that multiplying three negative numbers results in a negative number was key in calculating \(-\sqrt[3]{-1}\) and simplifying it to a positive result: \(1\).
Other exercises in this chapter
Problem 30
Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ \frac{1}{3^{-2}} $$
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Simplify the expression. Assume that all variables are positive. $$ \sqrt{16 x^{4} y^{6}} $$
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Use grouping to factor the polynomial. \(2 x^{4}-5 x^{3}+10 x-25\)
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Simplify. $$ \frac{3}{4}-\frac{7}{12} $$
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