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Chapter 7: Problem 48

Find each root. $$ \sqrt[3]{0.125} $$

Short Answer

Expert verified
The cube root of 0.125 is \( \frac{1}{2} \).

Step by step solution

01

Understand the Problem

The problem requires finding the cube root of 0.125. This means finding a number which, when multiplied by itself three times, equals 0.125.
02

Set Up the Equation

Let the cube root of 0.125 be represented by the variable, say, x. Therefore, we can set up the equation as follows: \[ x^3 = 0.125 \]
03

Solve for x

To find the value of x, we need to find the cube root of 0.125. Recall that \( 0.125 = \frac{1}{8} \). Since \( 8 = 2^3 \), \( \frac{1}{8} = \frac{1}{2^3} \). Therefore, \[ \ x^3 = \left(\frac{1}{2}\right)^3 \ x = \frac{1}{2} \]

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Key Concepts

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

cube root calculation
Cube root calculation involves finding a number that, when multiplied by itself three times, gives the original number. For instance, the cube root of 0.125 is a number that satisfies the equation \ \(x^3 = 0.125\ \). Since \ \(0.125\ \) can be expressed as \ \(\frac{1}{8}\ \), and knowing that \ \(8 = 2^3\ \), we get \ \(\frac{1}{8} = \frac{1}{2^3}\ \). Therefore, \ \(x = \frac{1}{2}\ \).
Knowing how to perform cube root calculations helps in various mathematical problems, especially in algebra and geometry.
exponents
Exponents represent repeated multiplication of a number by itself. In the equation \ \(x^3 = 0.125\ \), the exponent 3 in \ \(x^3\ \) indicates that x is multiplied by itself three times. Exponents can simplify expressions and calculations. For example:
  • \ \(2^3 = 2 \times 2 \times 2 = 8\ \).
  • \ \(10^2 = 10 \times 10 = 100\ \).
Understanding exponents is essential for solving polynomial equations and working with scientific notation.
fractions in algebra
Fractions involve a numerator and a denominator. They represent a part of a whole. In algebra, fractions can complicate equations but are solvable with the right approach. For example, in the given problem, we have \ \(\frac{1}{8}\ \) expressed as \ \(\frac{1}{2^3}\ \). Recognizing equivalent fractions and simplifying them is crucial.
It is helpful to remember rules like:
  • \ \(\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\ \),
  • multiplying both numerator and denominator by the same number does not change the fraction鈥檚 value.
solving equations
Solving equations requires finding the value(s) of the variable(s) that make the equation true. In the exercise, we have \ \(x^3 = 0.125\ \). We first converted 0.125 into \ \(\frac{1}{8}\ \) and recognized it as \ \(\frac{1}{2^3}\ \). This led us to solve for x by taking the cube root of both sides, giving us \ \(x = \frac{1}{2}\ \).
Equation-solving techniques are fundamental for algebra, calculus, and beyond. They allow us to discover unknown values and understand relationships between variables.

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