Problem 3 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ 16^{-3 / 4} $$

Step-by-Step Solution

Answer

The evaluation of \(16^{-3/4}\) is \( \frac{1}{8} \) or 0.125.

Step 1: Understand the Expression

The expression to evaluate is \(16^{-3/4}\). This is a power with a negative exponent and a fractional exponent.

Step 2: Apply the Negative Exponent Rule

According to the negative exponent rule, \(a^{-n} = \frac{1}{a^n}\). Thus, \(16^{-3/4} = \frac{1}{16^{3/4}}\).

Step 3: Convert the Base Expression to a Radical

The expression \(16^{3/4}\) can be rewritten as \((16^{1/4})^3\) because raising to a fractional power \(b/a\) means taking the \(a\)-th root and then raising to the \(b\)-th power. So, \(16^{3/4} = (16^{1/4})^3\).

Step 4: Simplify the Root

Find \(16^{1/4}\), which is the fourth root of 16. Since \(16 = 2^4\), the fourth root of 16 is 2. So, \(16^{1/4} = 2\).

Step 5: Raise to the Power

Now raise the result to the third power: \((16^{1/4})^3 = 2^3 = 8\). So, \(16^{3/4} = 8\).

Step 6: Solve the Original Expression

Substitute back into the expression from step 2: \(\frac{1}{16^{3/4}} = \frac{1}{8}\).

Step 7: Simplify the Result

The expression evaluates to \(\frac{1}{8}\), which can also be expressed as 0.125 in decimal form.

Key Concepts

Fractional Exponents

Fractional exponents are a way to represent powers and roots in a single expression. Fractional exponents look like fractions, where the numerator represents the power and the denominator represents the root. For example, in the expression \(16^{-3/4}\), the exponent \(-3/4\) signifies taking the fourth root (because of the denominator \(4\)) and then raising it to the power of \(-3\) (because of the numerator \(3\)).
Understanding fractional exponents is crucial for simplifying expressions and solving equations that involve roots and powers combined together.

When you encounter an expression like \(b/a\), remember to think of it as: take the \(a\)-th root and then raise it to the \(b\)-th power.
For example:

• \(16^{1/4}\) means "the fourth root of 16"
• Then, raising this result to the power of 3 gives \((16^{1/4})^3\)

Using this understanding makes it easier to manage complex calculations involving roots and powers together.

Exponent Rules

Exponent rules are foundational to simplifying and solving expressions involving powers. They help us rewrite expressions in more convenient forms to solve them easily. Here are some key rules:

• Negative Exponent Rule: For any non-zero number \(a\), \(a^{-n} = \frac{1}{a^n}\). This means you can transform negative exponents into fractions.
• Power of a Power Rule: \((a^m)^n = a^{m \times n}\). This means if you raise an exponent to another exponent, you multiply the exponents.
• Product of Powers Rule: \(a^m \times a^n = a^{m+n}\). This indicates adding the exponents when multiplying same bases.
• Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\). This indicates subtracting the exponents when dividing same bases.

Let's see these rules in action in our example. For \(16^{-3/4}\), the negative exponent rule helps transform it to \(\frac{1}{16^{3/4}}\). Understanding these rules helps you manipulate expressions easily and solve them confidently.

Radical Expressions

Radical expressions involve roots such as square roots, cube roots, fourth roots, and so on. They are usually expressed using the radical sign \(\sqrt{}\) or can be rewritten using fractional exponents to make calculations more convenient.

In our exercise, the expression \(16^{3/4}\) is equivalent to \((16^{1/4})^3\). Here, \(16^{1/4}\) is a radical expression that means the fourth root of 16. Finding the fourth root of 16 gives us \(2\) because \(2^4 = 16\).

By converting radical expressions into fractional exponents, and vice versa, we gain flexibility in solving mathematical problems. For instance, following these steps helped us compute \(16^{3/4}\) more easily by understanding it as \(2^3 = 8\). Understanding both approaches gives a stronger grip on handling complex calculations smoothly and effectively.