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Chapter 0: Problem 25

Simplify the expression. \(\sqrt[3]{-\frac{1}{8}}\)

Short Answer

Expert verified
-\frac{1}{2}

Step by step solution

01

Identify the components

Identify the components inside the radical. Here, the expression inside the cube root is \(-\frac{1}{8} \).
02

Separate negative sign

Recall that the cube root of a negative number is the negative of the cube root of the absolute value of the number. Thus, \(\root 3 \left(-\frac{1}{8}\right) = -\root 3 \left(\frac{1}{8}\right)\).
03

Simplify the fraction inside the cube root

Simplify the fraction inside the cube root. The cube root of \(\frac{1}{8}\) is \(\frac{1}{2}\) because \((\frac{1}{2})^3 = \frac{1}{8}\).
04

Combine the negative sign

Combine the negative sign with the cube root result from Step 3. Therefore, \(-\root 3 \left(\frac{1}{8}\right) = -\frac{1}{2}\).

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Key Concepts

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

Cube Roots
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Think of it like this: \ \(\text{cube root of x} = y\) means \(y^3 = x\).
For example, \(\root 3 \text {27}\) is 3, because \(3^3 = 27\).
Cube roots are different from square roots because they work with both positive and negative numbers.
If you take the cube root of a negative number, the result will also be negative.
For instance, the cube root of -8 is -2 because \((-2)^3 = -8\).
Simplifying Fractions
Simplifying fractions means to reduce them to their simplest form.
This makes the fraction as easy to work with as possible.
For example, \(\frac{2}{4}\) can be simplified to \(\frac{1}{2}\) by dividing both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD), which is 2.
In the problem, we simplified \(\frac{1}{8}\) by taking the cube root of both parts.
The cube root of 1 is just 1 (since \(1^3\) is 1) and the cube root of 8 is 2 (since \(2^3\) is 8).
This gave us the simplified fraction \(\frac{1}{2}\).
Negative Numbers
Negative numbers are numbers less than zero.
They are often used to represent values below a chosen reference point.
For example, -5 degrees could mean 5 degrees below zero.
When dealing with cube roots, remember that the cube root of a negative number is also negative.
This is because multiplying three negative numbers together still gives a negative product.
So, the cube root of -8 is -2 because \((-2)^3 = -8\).
Always be careful with signs when working through problems involving negatives.
They can change the outcome of your calculations.

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Most popular questions from this chapter

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (2 v-\sqrt{17})^{2} $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ 5 \sqrt{7}(3 \sqrt{7}-2 \sqrt{5}+3) $$

Use a calculator to approximate the expression to 2 decimal places. $$\frac{-3+5 \sqrt{2}}{7}$$

Simplify each expression. Assume that \(m\) and \(n\) are integers and that \(x\) and \(y\) are nonzero real numbers. \(\frac{x^{4 m-3} y^{5 n+7}}{x^{m-7} y^{3 n+2}}\)

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (5 \sqrt{z}-4)(3 \sqrt{z}+6) $$
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