Problem 10 - Precalculus: Functions and Graphs

Question

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 is equivalent to \( \frac{25}{1} \).

Step 1: Convert to Fraction

To begin, rewrite 0.008 as a fraction. Since 0.008 is equivalent to 8 thousandths, it can be expressed as \( \frac{8}{1000} \).

Step 2: Simplify the Fraction

Simplify the fraction \( \frac{8}{1000} \) by finding the greatest common divisor of 8 and 1000, which is 8. Dividing numerator and denominator by 8 gives \( \frac{1}{125} \).

Step 3: Apply the Exponent

Apply the exponent \( -\frac{2}{3} \) to the simplified fraction \( \frac{1}{125} \). Thus, you have \( \left( \frac{1}{125} \right)^{-\frac{2}{3}} \).

Step 4: Convert Negative Exponent to Positive

The negative exponent indicates the reciprocal should be taken. Hence: \( \left( \frac{1}{125} \right)^{-\frac{2}{3}} = \left( 125 \right)^{\frac{2}{3}} \).

Step 5: Evaluate the Fractional Exponent

To evaluate \( 125^{\frac{2}{3}} \), first find the cube root of 125, which is 5. Then, square the result: \( 5^2 = 25 \).

Step 6: Express Result as a Fraction

The final result is 25, which can be expressed as \( \frac{25}{1} \), where \( 25 \) and \( 1 \) are integers.

Key Concepts

Negative Exponents

When dealing with negative exponents, it's key to remember that they represent the reciprocal of the base raised to the positive of that exponent. This can be a tricky concept at first, but it simplifies many expressions, especially when dealing with fractions.
For example, if you have a number raised to \(-n\), like \(a^{-n}\), you can rewrite it as \(\frac{1}{a^n}\).
This process of converting a negative exponent into a fraction is a powerful tool in algebra, helping to simplify calculations and understand more complex expressions.

• Always remember: a negative exponent does not mean a negative number; it indicates a fraction.
• Converting negative exponents helps manage and simplify expressions, especially when combined with other operations.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is incredibly useful in simplifying fractions. It is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder.
To find the GCD of two numbers, such as 8 and 1000, you can use various methods. One approach is to list out the factors of each number and find the greatest one they have in common.
For 8 and 1000, the GCD is 8 because 8 is the largest number that evenly divides both.
Simplifying using the GCD produces smaller numbers, making further operations easier and more manageable.

• Using the GCD simplifies fractions quickly and efficiently.
• Keep in mind that simplifying fractions makes them easier to work with in calculations and comparisons.

Cube Roots

Cube roots are fundamental in dealing with fractional exponents. The cube root of a number \(x\) is a value \(y\) such that \(y^3 = x\).
For example, when evaluating an expression like \(125^{\frac{2}{3}}\),
we first identify that \(\frac{1}{3}\) in the exponent indicates a cube root.
Thus, the cube root of 125 is 5 because \(5 \times 5 \times 5 = 125\). After finding the cube root, you often need to raise the result to the remaining power, as in this case with \(5^2\).

• Cube roots and other roots are critical tools for solving equations and simplifying expressions.
• Always ensure you fully resolve any roots before proceeding with other calculations.

Fraction Simplification

Simplifying fractions is about reducing them to their simplest form. This usually involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
Consider the fraction \(\frac{8}{1000}\). Using the GCD, which we've calculated as 8, we divide both the numerator and the denominator by 8. This gives us \(\frac{1}{125}\), a much simpler fraction to work with.
Fraction simplification is a key component in algebra and calculus, enabling easier handling of expressions and equations.

• Always check for the GCD to simplify fractions; this helps in better handling and understanding of mathematical problems.
• Simplified fractions make it easier to add, subtract, multiply, or divide values across different mathematical contexts.