Problem 67 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ \frac{4^{2 / 3}}{4^{1 / 2}} $$

Step-by-Step Solution

Answer

The expression evaluates to approximately 1.26.

Step 1: Apply the Quotient of Powers Property

The quotient of powers property states that \( \frac{a^m}{a^n} = a^{m-n} \). In the given expression \( \frac{4^{2/3}}{4^{1/2}} \), we can apply this property. So, we get:\[ a^{\frac{2}{3} - \frac{1}{2}} \]

Step 2: Subtract the Exponents

Subtract \( \frac{1}{2} \) from \( \frac{2}{3} \). To do this, find a common denominator for the fractions, which is 6. Therefore, \( \frac{2}{3} = \frac{4}{6} \) and \( \frac{1}{2} = \frac{3}{6} \). Subtraction gives us:\[ \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \]

Step 3: Simplify the Expression

Now the expression simplifies to \( 4^{\frac{1}{6}} \).

Step 4: Evaluate the Expression

To find \( 4^{\frac{1}{6}} \) numerically, use a calculator. Calculate the sixth root of 4, which approximates to:\[ 4^{\frac{1}{6}} \approx 1.26 \](rounded to the nearest hundredth).

Key Concepts

Quotient of Powers

The quotient of powers is a useful property when dealing with expressions involving the division of like bases. Specifically, the rule states:

• Given two exponents with the same base, such as \( \frac{a^m}{a^n} \), the operation becomes \( a^{m-n} \).

This means we subtract the exponent in the denominator from the exponent in the numerator. In the example where we have \( \frac{4^{2/3}}{4^{1/2}} \), the base \(4\) is the same in both the numerator and denominator.

By using the quotient of powers property, we rewrite the expression as \( 4^{(2/3)-(1/2)} \). Thus, solving exercises with this property not only simplifies expressions but also prepares them for further computation, making the next steps easier.

Exponentiation

Exponentiation refers to the mathematical operation involving numbers called bases and exponents. It is expressed as \( a^n \), which means \( a \) is multiplied by itself \( n \) times. However, exponents don't only have to be whole numbers—fractions can be used as well.

• Fractional exponents like \( a^{1/2} \) correspond to operations such as taking the square root of \( a \).
• Similarly, \( a^{1/3} \) is equivalent to the cube root of \( a \), and \( a^{n/m} \) is the \( m^{th} \) root of \( a \) raised to the \( n^{th} \) power.

In our example, embracing the concept of fractional exponents allows us to transform the problem \( 4^{2/3} \) into an easier form to compute. Understanding fractional exponents expands our mathematical toolkit, letting us handle a broader set of mathematical problems.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, where they are easier to interpret or solve.

• Using properties like the quotient of powers helps in minimizing the complexity of an algebraic problem.
• Finding common denominators is crucial when subtracting fractions, as seen when transitioning \( 4^{(2/3)-(1/2)} \) into \( 4^{1/6} \).

In this step-by-step example, simplifying the expression led us to \( 4^{1/6} \), a much simpler form that could be directly evaluated using numerical approximations, such as with calculators. Simplifying not only helps in manual calculations, but is essential for real-world applications, where easier-to-compute numbers matter.