Problem 71 - Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ \left(-\frac{27}{8}\right)^{-4 / 3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{16}{81}\).

Step 1: Apply Negative Exponent Rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^{n}}\). Apply this rule to the expression \(\left(-\frac{27}{8}\right)^{-4/3}\):\[\left(-\frac{27}{8}\right)^{-4/3} = \frac{1}{\left(-\frac{27}{8}\right)^{4/3}}\]

Step 2: Rewrite the Base as a Fraction Power

Recognize that raising a fraction to a power \(\left(\frac{a}{b}\right)^n\) equals \(\frac{a^n}{b^n}\). Apply this to the expression \(\left(\frac{1}{\left(-\frac{27}{8}\right)^{4/3}}\right)\):\[\frac{1}{\left(-\frac{27}{8}\right)^{4/3}} = \frac{1}{\left((-27)^{4/3} \div 8^{4/3}\right)} = \frac{8^{4/3}}{(-27)^{4/3}}\]

Step 3: Simplify Each Component Separately

Simplify \(8^{4/3}\) and \((-27)^{4/3}\). Use the following reasoning: \(a^{m/n} = (a^{1/n})^m\):- \(8^{4/3} = (8^{1/3})^4 = (2)^4 = 16\), because \(8^{1/3} = 2\).- \((-27)^{4/3} = ((-27)^{1/3})^4\). Since \((-27)^{1/3} = -3\), then \((-27)^{1/3}\) remains valid for positive evaluation of real cube root.- Thus, \((-3)^4 = 81\).

Step 4: Calculate the Final Fraction

Combine the results from Step 3 to express the final calculation:\[\frac{8^{4/3}}{(-27)^{4/3}} = \frac{16}{81}\]

Key Concepts

Negative Exponent Rule

When faced with a negative exponent, it can initially seem daunting, but the negative exponent rule simplifies things. This rule states that any base with a negative exponent can be transformed by taking the reciprocal of the base and making the exponent positive.
For example, if you encounter an expression like \((-\frac{27}{8})^{-4/3}\), you can change it using the rule:

• Switch the fraction and make the exponent positive: \((-\frac{27}{8})^{-4/3} = \frac{1}{(-\frac{27}{8})^{4/3}}\)

By applying this rule, we prepare the expression for easier simplification later on, as it allows us to work with positive exponents instead of negative ones.

Fractional Exponents

Fractional exponents, also known as rational exponents, can be puzzling, but they actually represent roots and powers rolled into one simple expression. When you see an exponent like \(a^{m/n}\), it means you are dealing with an n-th root raised to the m-th power.
Let's break it down:

• \(a^{1/n}\) is the n-th root of a.
• \((a^{1/n})^m = a^{m/n}\), which means you first find the n-th root and then raise it to the m-th power.

In our example, we had to simplify \((-27)^{4/3}\), which involves:

• Finding the cube root of -27, giving -3.
• Raising -3 to the 4th power, resulting in 81.

This understanding helps unravel the simplicity within seemingly complicated expressions.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form by ensuring the numerator and the denominator have no common factors other than 1. This methodology becomes handy when expressions involve fractional bases or exponents.
In our solved problem, after evaluating the expression with fractional exponents, we have to simplify the fraction:

• \(\frac{8^{4/3}}{(-27)^{4/3}}\)
• Simplified separately, \(8^{4/3} = 16\) and \((-27)^{4/3} = 81\).
• Thus, the overall fraction becomes \(\frac{16}{81}\).

Since 16 and 81 do not share any factors other than 1, this fraction is already in its simplest form, illustrating the power and continuity of simplification techniques.