Problem 7 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ \left(9^{3 / 4}\right)^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 27.

Step 1: Understand the expression

The expression we need to evaluate is \( \left(9^{3/4}\right)^2 \). This means that we are first raising \(9\) to the power \(3/4\) and then raising the result to the power 2.

Step 2: Apply the power of a power rule

When you have an expression like \((a^m)^n\), you can apply the power of a power rule, which states that we multiply the exponents: \(a^{m \cdot n}\). Here, \(a = 9\), \(m = 3/4\), and \(n = 2\).

Step 3: Calculate the new exponent

Multiply the exponents: \(\frac{3}{4} \times 2 = \frac{3 \times 2}{4} = \frac{6}{4} = \frac{3}{2}\). So, \(\left(9^{3/4}\right)^2 = 9^{3/2}\).

Step 4: Rewrite the expression with the new exponent

Now, you need to evaluate \(9^{3/2}\). This is the same as \((9^{1/2})^3\).

Step 5: Evaluate \(9^{1/2}\)

\(9^{1/2}\) means the square root of 9. Since the square root of 9 is 3, \(9^{1/2} = 3\).

Step 6: Evaluate \((3)^3\)

Now, raise 3 to the 3rd power: \(3^3 = 3 \times 3 \times 3 = 27\).

Step 7: Write the final answer

So, \(\left(9^{3/4}\right)^{2} = 27\).

Key Concepts

Power of a Power Rule

The power of a power rule is an essential concept when working with exponents. It simplifies the process of dealing with repeated exponentiation. When a number is raised to an exponent and then the result is raised to another exponent, the exponents can be multiplied together. This principle is expressed through the equation

• \( (a^m)^n = a^{m \cdot n} \)

In the context of our problem, where we have \( (9^{3/4})^2 \), the rule allows us to multiply the exponent \( \frac{3}{4} \) by 2, resulting in \( 9^{3/2} \).
This method streamlines the calculation by consolidating multiple steps into one efficient operation. By using this rule, we avoid more complex calculations and maintain the integrity of the expression.

Radicals

Radicals are another form of expression that can often look complex at first. They are symbols that denote roots of numbers, with the most common being the square root, expressed as \( \sqrt{} \).

• For example, the square root of 9 is written as \( \sqrt{9} = 3 \).
• Radicals can also be expressed in terms of exponents.

The relationship between radicals and exponents becomes clear when you consider the problem at hand:\( 9^{1/2} \) is equivalent to the square root of 9.
Understanding this relationship helps in transforming an expression involving radicals into a form that might be easier to manipulate or compute, depending on the context of the problem. This transformation between radical notation and exponential form broadens our ability to solve such expressions efficiently.

Rational Exponents

Rational exponents are a generalization of integral exponents that allow for greater flexibility and expression clarity. When dealing with rational exponents, the numerator indicates the power, and the denominator indicates the root. This means that

• \( a^{p/q} \) represents the \( q \)th root of \( a \) raised to the \( p \)th power.
• For example, \( 9^{3/2} \) translates to taking the square root of 9 (as seen previously with \( 9^{1/2} = 3 \)) and then raising the result to the power of 3.

This exponentiation format is highly efficient for solving complex expressions by reducing the number of steps required to arrive at a solution. In our case, by directly translating the exponents in this form and applying the necessary mathematical steps, we derived that \( 9^{3/2} = 3^3 = 27 \).
Employing rational exponents makes it easier to understand and handle expressions by using arithmetic and algebraic operations effectively.