Problem 31 - Intermediate Algebra

Question

Write with positive exponents. Simplify if possible. $$ (-64)^{-2 / 3} $$

Step-by-Step Solution

Answer

The expression \((-64)^{-2/3}\) simplifies to \(\frac{1}{16}\).

Step 1: Understand Negative Exponents

The expression \((-64)^{-2/3}\) contains a negative exponent. Remember, a negative exponent means taking the reciprocal of the base. Therefore, one way to rewrite the expression is by writing it as \(\frac{1}{(-64)^{2/3}}\).

Step 2: Simplify the Fractional Exponent

The fractional exponent \(\frac{2}{3}\) indicates a root and a power. Generally, \(b^{m/n} = (\sqrt[n]{b})^m\). Apply this to \(((-64)^{2/3})\), which implies \(((\sqrt[3]{-64})^2)\), involving finding the cube root first, then squaring it.

Step 3: Calculate the Cube Root

Calculate the cube root of \(-64\) which is \(-4\), because \((-4)^3 = -64\). Thus, \(\sqrt[3]{-64} = -4\).

Step 4: Square the Result

Now, square the result from Step 3: \((-4)^2 = 16.\) Consequently, the expression from Step 2 becomes \(\left((-64)^{2/3}\right) = 16\).

Step 5: Write Reciprocal for Positive Exponent

Since we initially flipped the base to make the exponent positive, we finally use the reciprocal from Step 1 to arrive at \((-64)^{-2/3} = \frac{1}{16}\).

Key Concepts

Fractional Exponents

Fractional exponents can be confusing at first, but they are actually very handy. In general, a fractional exponent, such as \( b^{m/n} \), represents two operations: taking the \( n \)-th root of \( b \) and then raising that result to the \( m \)-th power. This two-step operation gives us the flexibility to break down complex calculations.
For instance, with the expression \( (-64)^{-2/3} \), the fractional part \( \frac{2}{3} \) means two operations are involved: cubic rooting \((-64)\) and squaring the result of the root.

• The numerator of the fraction indicates the power.
• The denominator tells us which root to take.

Understanding this can make working with fractional exponents much simpler.

Simplifying Expressions

Simplifying expressions is crucial to make calculations easier and faster. It involves reducing an expression to its most concise form, which is often necessary when dealing with complex numbers or an equation.
In our example \( (-64)^{-2/3} \), simplifying involved recognizing that the exponent can be rewritten in terms of a reciprocal when negative. Additionally, we used the properties of fractional exponents to break down the problem into simpler parts—calculating a cube root and then squaring.

• Identify opportunities to rewrite negative or fractional exponents.
• Break down the operations step by step.

By simplifying, you get a clearer, often more manageable view of what you're working with.

Reciprocal

A reciprocal is an important concept, especially when dealing with negative exponents. It signifies "flipping" a number, essentially turning a fraction upside down. For a number \( a \), its reciprocal is \( \frac{1}{a} \).
In the context of exponents, a negative exponent implies taking the reciprocal of the base. So, if you see \( (-64)^{-2/3} \), this instructs us first to find the reciprocal of \( (-64)^{2/3} \).

• Negative exponent? Think reciprocal.
• Flip the base structure for negative exponents.

This simplification often converts hard problems into something more tangible and approachable.

Cube Root

The cube root is a specific root corresponding to the exponent 1/3. It helps simplify expressions where something is raised to a power of three. The cube root, \( \sqrt[3]{b} \), is the number that, when cubed, results in the original \( b \).
For example, \( \sqrt[3]{-64} \) equals \(-4\) because \((-4)^3 = -64\). It's essential to understand roots to solve problems with fractional exponents effectively.

• The cube root of a number is key when the fraction exponent base is 3.
• Always check that cubing the result gives back your original value.

Getting comfortable with cube roots allows you to handle expressions that seem difficult at first glance, making them more manageable.