Problem 9 - 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 \((-0.008)^{2/3}\) simplifies to \(1/4\).

Step 1: Rationalize the Exponent

The given expression is \((-0.008)^{2/3}\). To simplify the calculation, first recognize \(-0.008\) as a power of \(-2)\) and convert the fractional exponent to roots. Write \(-0.008 = -(2^{-3}) = (-2^{-3})\).

Step 2: Apply the Exponent to the Power

Apply the fractional exponent \(2/3\) to each component: \((-2^{-3})^{2/3} = (-1 imes 2^{-3})^{2/3}\). The formula reads as \((-1)^{2/3} imes (2^{-3})^{2/3}\).

Step 3: Simplify the Expression

Simplify each part of the expression. The part \((-1)^{2/3}\) simplifies to \(1\) since any real number raised to an even power is positive. The power \((2^{-3})^{2/3}\) becomes \(2^{-2}\).

Step 4: Evaluate the Negative Power

Calculate \(2^{-2}\), which equals \(1/2^2 = 1/4\). This reflects the conversion of a negative power into a fraction.

Step 5: Combine Results

Combine the results from the simplification: \((-1)^{2/3} imes 2^{-2} = 1 imes 1/4 = 1/4\). Thus, the expression simplifies to \(1/4\).

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, are a unique way of expressing powers and roots in a single expression. For example, if you see an exponent expressed as a fraction like \(\frac{2}{3}\), it combines the concepts of both multiplication (the numerator, 2) and roots (the denominator, 3).
This special numerical notation allows you to express a root and a power at the same time, such as \((-0.008)^{2/3}\) in our exercise.
Here's what each part of \((-0.008)^{2/3}\) means:

• The numerator (2) indicates the power to which you are raising the base number.
• The denominator (3) indicates that you are taking the cube root of the raised power.

In general, an expression \(x^{m/n}\) can be interpreted as \(\sqrt[n]{x^m}\) or equivalently \((\sqrt[n]{x})^m\). This allows you to manipulate roots and powers in flexible ways. Recognizing this pattern is crucial when dealing with complex numbers or expressions involving negative bases.

Exponent Rules

Exponent rules are the guiding principles that let you manipulate and simplify expressions involving powers with ease.
They are invaluable, especially when working with fractional exponents. Here are a few essential rules you should know:

• Product of Powers Rule: If you multiply two numbers with the same base, you add their exponents. For example, \(x^a \times x^b = x^{a+b}\).
• Quotient of Powers Rule: If you divide two numbers with the same base, you subtract their exponents. For example, \(x^a / x^b = x^{a-b}\).
• Power of a Power Rule: If you raise an exponent to another exponent, you multiply the exponents. For example, \((x^a)^b = x^{a\times b}\).

In our exercise, we use these rules to simplify \(\left((-2)^{-3}\right)^{2/3}\) by employing the power of a power rule. This results in \((-2)^{-2}\), which simplifies further to a more manageable expression. Keep these rules handy, as they are key to working through any problem involving exponents, whether integers or fractions.

Roots and Powers

Roots and powers are intertwined concepts in mathematics, particularly when dealing with fractional exponents.
Understanding them is crucial for simplifying expressions like the one in this exercise. To break it down, remember:

• Powers: Raising a number to a power means multiplying it by itself a certain number of times.
• Roots: Taking the root of a number is the opposite of raising it to a power; it essentially "undoes" the power.

Fractions like \(\frac{2}{3}\) blend these concepts, making them a powerful tool for expressing complex operations simply. In the given exercise, we see
\((-0.008)^{2/3}\), which involves taking the cube root of \(-0.008\) and then squaring that result.
This two-step process converts complex multiplication into straightforward arithmetic.
By clarifying these roots and powers, you transform difficult calculations into a series of simpler tasks, making seemingly challenging problems far more approachable.