Problem 19 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$\left(\frac{27}{64}\right)^{-4 / 3}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{256}{81} \).

Step 1: Apply the Negative Exponent Rule

The Negative Exponent Rule states that for any non-zero number \( a \), \( a^{-m} = \frac{1}{a^m} \). Applying this to our expression, we get: \[ \left( \frac{27}{64} \right)^{-\frac{4}{3}} = \frac{1}{\left( \frac{27}{64} \right)^{\frac{4}{3}}} \]

Step 2: Simplify Inside the Exponent

To simplify \( \left( \frac{27}{64} \right)^{\frac{4}{3}} \), we first find the cube root of the base fraction and then raise it to the fourth power. Begin by evaluating the cube roots: \[ \sqrt[3]{\frac{27}{64}} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4} \]

Step 3: Raise the Result to the Fourth Power

Now, raise the result from Step 2 to the fourth power: \(\left(\frac{3}{4}\right)^{4}=\frac{3^4}{4^4}\) Calculate this: \[\frac{3^4}{4^4} = \frac{81}{256}\]

Step 4: Invert the Result from Step 3

Since we initially applied the negative exponent rule in Step 1, we now take the reciprocal of our result from Step 3: \[ \frac{1}{\left( \frac{81}{256} \right)} = \frac{256}{81} \]

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest form so that the numerator and denominator have no common factors other than 1. To do this, follow a few simple steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The result is your simplified fraction.

Let's look at an example. Consider the fraction \( \frac{27}{64} \). It's already in its simplest form because 27 and 64 have no common factors other than 1. However, when fractions need simplification during exponent operations, it's always beneficial to simplify first if possible, making calculations straightforward and reducing the chances of error.
Additionally, operating with simpler numbers generally makes the processes like finding roots or powers easier to handle.

Cube Root

The cube root is used to determine which number multiplied by itself three times will give the original number. Basically, for a number \( a \), its cube root is \( a^{1/3} \). Assuming we have a fraction, like \( \frac{27}{64} \), we can find its cube root by separately taking the cube root of the numerator and the cube root of the denominator:

• The cube root of 27 is 3, because \( 3^3 = 27 \).
• The cube root of 64 is 4, because \( 4^3 = 64 \).

Thus, the cube root of \( \frac{27}{64} \) is \( \frac{3}{4} \). This process helps in restructuring fractions into more manageable numbers for further calculations, such as raising to higher powers.

Raising to a Power

Raising a number to a power involves multiplying the number by itself a specified number of times, noted as an exponent. For example, raising \( \frac{3}{4} \) to the fourth power means multiplying \( \frac{3}{4} \) by itself four times:

• \( \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \)

This operation can be simplified by raising both the numerator and the denominator to the exponent individually:

• \( 3^4 = 81 \)
• \( 4^4 = 256 \)

So, \( \left(\frac{3}{4}\right)^4 = \frac{81}{256}\). This method ensures precise computation when dealing with fractions, supporting our step-by-step approach to expression simplification.

Exponentiation Steps

Exponentiation processes require a few logical steps to ease understanding and correctness, particularly when dealing with fractions and complicated exponents. In the expression \( \left(\frac{27}{64}\right)^{-\frac{4}{3}} \), these steps follow a systematic approach:

• First, apply the negative exponent rule: converting \( a^{-m} \) to \( \frac{1}{a^m} \).
• Next, simplify where possible. In our case, start with finding the cube root of the fraction.
• Then, raise the simplified base to the indicated power. This means handling exponents like \( \left(\frac{3}{4}\right)^4 \).
• Finally, apply any inverses as required by negative exponents initially set in the expression.

Following these consecutive steps helps demystify complex expressions, underscoring clarity and correctness.