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Magazine Chapter 1: Problem 9
Express the number in the form a/b, where \(a\) and \(b\) are integers. $$(-0.008)^{2 / 3}$$
Short Answer
Expert verified
The number is expressed as \(\frac{1}{25}\).
Step by step solution
01
Understand the Problem
We are given a negative decimal number, \(-0.008\), raised to the power of \(2/3\). We need to express this number in the form \(a/b\), where both \(a\) and \(b\) are integers.
02
Convert Decimal to Fraction
Convert \(-0.008\) to a fraction. The decimal \(-0.008\) can be written as \(-\frac{8}{1000}\). This fraction can be simplified by dividing both the numerator and the denominator by 8, which results in \(-\frac{1}{125}\).
03
Simplify the Power Expression
We have \(\left(-\frac{1}{125}\right)^{2/3}\). This represents the cube root of \(\left(-\frac{1}{125}\right)^2\).
04
Compute the Square of the Fraction
First, calculate \(\left(-\frac{1}{125}\right)^2\). Squaring the fraction gives \(\frac{1}{15625}\).
05
Find the Cube Root
Finally, find the cube root of \(\frac{1}{15625}\). The cube root of \(\frac{1}{15625}\) is \(\frac{1}{25}\) because \(25^3 = 15625\).
06
Present the Result
We obtain \(\left(-0.008\right)^{2/3} = \frac{1}{25}\). Therefore, \(a = 1\) and \(b = 25\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Converting Decimals to Fractions
Converting decimals to fractions is a fundamental skill that helps us express numbers in a different form. To start converting, identify the place value of the last digit in the decimal. For example, in the decimal \(-0.008\), the '8' is in the thousandths place. Thus, \(-0.008 = -\frac{8}{1000}\).
- Count the number of decimal places: Three places means dividing by 1000 (10^3).
- Write the decimal as a fraction using this denominator.
- Simplify the fraction by finding the greatest common divisor.
- In \(-\frac{8}{1000}\), dividing both parts by 8 gives \(-\frac{1}{125}\).
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator share no common factors other than 1. Once you have a fraction, such as \(-\frac{8}{1000}\), turning it into \(-\frac{1}{125}\) is simplifying it.
- Determine the greatest common divisor (GCD) of the numerator and the denominator.
- Divide both by the GCD to reduce the fraction. Here, 8 is the GCD.
- Always check if further simplification is possible.
Cube Roots
Cube roots are a critical concept when dealing with rational exponents like \(\frac{2}{3}\). The cube root of a number is a value that, when multiplied by itself three times, gives the original number.Take \(\frac{1}{15625}\) as an example. To find the cube root,
- You are seeking a number \((x)\) such that \(x^3 = 15625\).
- Here, \((25^3 = 15625)\) shows that 25 is the cube root.
- The cube root of \(\frac{1}{15625}\) is \(\frac{1}{25}\).
Rational Exponents
Rational exponents extend the idea of powers to include fractions. When faced with an expression like \((-0.008)^{2/3}\), break it into manageable parts: exponents and roots.
- The numerator of the exponent (2) signifies squaring the base.
- The denominator (3) indicates taking the cube root following the power.
- This fractional power is equivalent to a combination of a square followed by a cube root.
