Problem 80 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{64 a^{\frac{1}{3}}}{16 a^{\frac{5}{9}}}\right)^{3}\)

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{64}{a^{\frac{2}{3}}} \).

Step 1: Distribute the Exponent

Apply the power of a power rule by distributing the exponent 3 to both the numerator and the denominator: \[ \left(\frac{64 a^{\frac{1}{3}}}{16 a^{\frac{5}{9}}}\right)^{3} = \frac{64^{3} \cdot \left(a^{\frac{1}{3}}\right)^{3}}{16^{3} \cdot \left(a^{\frac{5}{9}}\right)^{3}} \]

Step 2: Simplify Powers of Numbers

Calculate the powers of 64 and 16: \[ 64^{3} = (4^{3})^{3} = 4^{9} \] \[ 16^{3} = (4^{2})^{3} = 4^{6} \] Replace these in the fraction: \[ \frac{4^{9} \cdot \left(a^{\frac{1}{3}}\right)^{3}}{4^{6} \cdot \left(a^{\frac{5}{9}}\right)^{3}} \]

Step 3: Simplify Powers of Variables

Apply the power of a power rule on the variable expressions: \[ \left(a^{\frac{1}{3}}\right)^{3} = a^{\frac{1}{3} \times 3} = a^{1} = a \] \[ \left(a^{\frac{5}{9}}\right)^{3} = a^{\frac{5}{9} \times 3} = a^{\frac{15}{9}} = a^{\frac{5}{3}} \] The fraction now becomes: \[ \frac{4^{9} \cdot a}{4^{6} \cdot a^{\frac{5}{3}}} \]

Step 4: Simplify the Fraction

Simplify the fraction by dividing the powers of 4: \[ \frac{4^{9}}{4^{6}} = 4^{9-6} = 4^{3} \] Apply the division rule for the powers of a: \[ \frac{a}{a^{\frac{5}{3}}} = a^{1-\frac{5}{3}} = a^{-\frac{2}{3}} \] Thus, the expression simplifies to: \[ 4^{3} \cdot a^{-\frac{2}{3}} \]

Step 5: Express Final Result Using Positive Exponents

Rewrite the expression to have only positive exponents: \[ 4^{3} \cdot \frac{1}{a^{\frac{2}{3}}} = \frac{4^{3}}{a^{\frac{2}{3}}} \] Since \[ 4^{3} = 64 \], the final result is: \[ \frac{64}{a^{\frac{2}{3}}} \]

Key Concepts

Power of a Power Rule

When dealing with expressions that have exponents raised to another exponent, the power of a power rule becomes extremely useful. The power of a power rule states that when you raise a power to another power, you multiply the exponents together. For example, if you have \(a^{m}\) raised to the power of n, it becomes \(a^{m \cdot n}\).

Let's take an example given by the exercise: \(\left(a^{\frac{1}{3}}\right)^{3}\). According to the power of a power rule, you multiply \frac{1}{3}\ by 3, which gives you 1. Therefore, \(\left(a^{\frac{1}{3}}\right)^{3} = a^{1} = a\).

Similarly, for \(\left(a^{\frac{5}{9}}\right)^{3}\), you multiply \(\frac{5}{9} \) by 3 to get \(a^{\frac{15}{9}}\), which simplifies to \(a^{\frac{5}{3}}\). This rule allows you to handle complex expressions simply by performing multiplication of the exponents.

Simplifying Expressions

Simplifying expressions involves reducing them into the simplest form possible by using mathematical rules and operations. This can involve combining like terms, using exponent rules, and performing arithmetic operations.

In our example, once the power of a power rule has been applied, the next step involves simplifying the numbers and variables separately. For instance, \(64^{3}\) and \(16^{3}\) were simplified by breaking them down into base powers of 4. So, \(64 = 4^{3}\) leads us to \(64^{3} = (4^{3})^{3} = 4^{9}\). Similarly, for \(16^{3}\), note that \(16 = 4^{2}\), resulting in \(16^{3} = (4^{2})^{3} = 4^{6}\).

After simplifying these term-wise, the next key task is the reduction of the fraction \(\frac{4^{9} \cdot a}{4^{6} \cdot a^{\frac{5}{3}}}\), which further entails applying the division rule of exponents. Simplifying involves performing simple subtraction of exponents based on like bases, such as \(4^{9-6}\) and \(a^{1-\frac{5}{3}}\), to arrive at the simplest form of the expression.

Positive Exponents

Exponents are a concise way to express repeated multiplication. However, expressions always look neater and are more mathematically standard when expressed using positive exponents. This can involve rewriting negative exponents by transforming them into fractions.

In our step-by-step problem solution, the final expression is \(4^{3} \cdot a^{-\frac{2}{3}}\). Here, \(a^{-\frac{2}{3}}\) is rewritten to use positive exponents by moving the term from the numerator to the denominator. This follows the rule: \(a^{-b} = \frac{1}{a^{b}}\).

Thus, changing \(a^{-\frac{2}{3}}\) to positive gives \(\frac{1}{a^{\frac{2}{3}}}\). Bringing it all together, with \(4^{3} = 64\), makes the expression \(\frac{64}{a^{\frac{2}{3}}}\), so that the result consists entirely of positive exponents—all terms in their simplest positive form.