Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: Express the number in the form \(a / b\), where \(a\) and \(b\) are integers. $$ (0.008)^{-2 / 3} $$

Step-by-Step Solution

Answer

The expression \( (0.008)^{-2/3} \) is equivalent to \( \frac{25}{1} \).

Step 1: Convert Decimal to Fraction

First, express the decimal number 0.008 as a fraction. This is equivalent to \( \frac{8}{1000} \) because there are three decimal places.

Step 2: Simplify the Fraction

Simplify the fraction \( \frac{8}{1000} \) by finding the greatest common divisor (GCD) of 8 and 1000. The GCD is 8, so \( \frac{8}{1000} \) simplifies to \( \frac{1}{125} \).

Step 3: Apply the Exponent to the Fraction

The expression \( (0.008)^{-2/3} \) is equivalent to \( \left( \frac{1}{125} \right)^{-2/3} \). When raising a fraction to a power, both the numerator and the denominator are raised to that power.

Step 4: Use Negative Exponent Rule

The negative exponent \( -\frac{2}{3} \) indicates taking the reciprocal of the base and then raising it to the positive exponent \( \frac{2}{3}\). So, \( \left( \frac{1}{125} \right)^{-2/3} = (125)^{2/3} \).

Step 5: Convert Radical Exponent to Root and Power

Recognize that the exponent \( \frac{2}{3} \) can be interpreted as the cube root of \( 125 \) squared. Calculate first the cube root of \( 125 \), which is 5, since \( 5^3 = 125 \).

Step 6: Compute the Power

Now, square 5. This gives \( 5^2 = 25 \).

Step 7: Express the Final Answer as Integer

Therefore, the expression has been simplified to 25, which is \( \frac{25}{1} \), confirming \( a = 25 \) and \( b = 1 \).

Key Concepts

Exponent Rules

Understanding exponent rules is essential when dealing with expressions like \((0.008)^{-2/3}\). An exponent indicates how many times you multiply a number by itself.
In this exercise, we need to manage both a fractional exponent and a negative exponent. With fractional exponents, like \(-2/3\), the numerator
represents the power, while the denominator represents the root. This means \((x)^{a/b}\) is equivalent to taking
the \(b\)-th root of \(x\) and then raising the result to the \(a\)-th power.

• Negative exponents, such as \(-2/3\), signify the reciprocal operation. So, \((x)^{-a}\) becomes \(1/(x^{a})\).
• In practical terms, this often involves converting negative exponents into a form
  that makes calculations or simplifications easier.

Recognizing these rules is fundamental when simplifying expressions, as they
allow us to break down complex exponential expressions into manageable pieces.

Simplifying Fractions

Simplifying fractions is a crucial step in managing algebraic expressions. It involves reducing fractions to their simplest terms by dividing
the numerator and the denominator by their greatest common divisor (GCD).
For example, the decimal 0.008 was converted into the fraction \(\frac{8}{1000}\).

• By identifying the GCD of 8 and 1000, which is 8, we were able to simplify
  \(\frac{8}{1000}\) to \(\frac{1}{125}\).
• This simplified fraction makes further calculations more straightforward
  and less error-prone, especially when applying exponent rules.

The process of simplifying fractions helps maintain the elegance and clarity of mathematical expressions,
ensuring calculations remain concise and accurate.

Negative Exponents

Negative exponents can seem challenging, but they follow a simple rule: indicate taking the reciprocal of the base and applying the positive exponent.
For instance, with \((0.008)^{-2/3}\), after converting 0.008 to \(\frac{1}{125}\), the rule implies we take the reciprocal: \((125)^{2/3}\).

• This is not just flipping the fraction; it has conceptual foundations in how
  inverse quantities of multiplication are handled in mathematics.
• Understanding negative exponents means recognizing they transform problems from division into multiplication.

Comprehending negative exponents is an important skill, as it allows us to
rearrange expressions efficiently, particularly when managing more complex calculus operations.

Radical Exponents

Radical exponents, or fractional exponents, are a way of expressing powers and roots in a single expression. In the case of \(\frac{2}{3}\), the expression can be broken down into taking the cube root first, then squaring the result.

• For instance, \((125)^{2/3}\) involves finding the cube root of 125, which is 5
  because \(5^3 = 125\).
• Then, raise the result, 5, to the power of 2, giving \(5^2 = 25\).

This method streamlines the process of working with powers and roots, allowing simplified calculations on otherwise unwieldy expressions and promoting a deeper understanding of how different mathematical operations are interconnected.