Problem 45 - Algebra A Combined Function

Question

Evaluate each exponential expression. $$ 16^{3 / 4} $$

Step-by-Step Solution

Answer

The value of \(16^{3/4}\) is 8.

Step 1: Understanding the Problem

We need to evaluate the exponential expression \(16^{3/4}\). This involves applying the concept of fractional exponents.

Step 2: Simplifying the Base

First, recognize that 16 can be rewritten as \(2^4\) because \(2 \times 2 \times 2 \times 2 = 16\). This helps simplify the expression.

Step 3: Applying the Fractional Exponent

By rewriting \(16^{3/4}\) using the base 2, we have: \((2^4)^{3/4}\). By the rule of exponents \((a^m)^n = a^{m \cdot n}\), the expression becomes \(2^{4 \times (3/4)}\).

Step 4: Calculating the Exponent

Calculate the exponent: \(4 \times \frac{3}{4} = 3\). The expression simplifies to \(2^3\).

Step 5: Solving the Final Expression

Compute \(2^3\) to get the final answer: \(2 \times 2 \times 2 = 8\).

Key Concepts

Fractional Exponents

Fractional exponents may seem intimidating at first, but they are really just a different way of expressing roots and powers. When we see an exponent written as a fraction, such as \(16^{3/4}\), it indicates both a root and a power.

• The denominator (bottom part) of the fraction tells you which root to take. In our example, "4" means the fourth root.
• The numerator (top part) tells you the power to which you'll raise the result. When we have "3/4" in the exponent, it means we first find the fourth root and then raise the outcome to the power of 3.

Using fractional exponents can make working with large numbers simpler and more intuitive. Instead of finding massive powers, fractional exponents allow us to break down calculations into more manageable steps.

Exponential Rules

When dealing with exponent expressions, several rules are useful for simplifying calculations. These rules help us manage expressions like \((a^m)^n = a^{m \cdot n}\).

• The power of a power rule: When you raise a power to another power, multiply the exponents. For example, \((2^4)^{3/4}\) becomes \(2^{4 \times (3/4)} = 2^3\).
• Product of powers rule: When multiplying two exponents with the same base, add the exponents, evident if we had something like \(a^m \times a^n = a^{m+n}\).
• Quotient of powers rule: When dividing two exponents with the same base, subtract the exponents. It looks like \(a^m / a^n = a^{m-n}\).

Understanding and applying these rules helps make complex problems simple and is essential for mastering exponential expressions.

Simplifying Expressions

Simplifying expressions can turn a complicated-looking mathematical problem into a simpler one. By organizing and reducing the parts of an expression, we achieve a clear and straightforward solution path.

• Rewrite complex bases: Convert bases into their simplest forms if possible, e.g., rewrite 16 as \(2^4\), as it makes calculations much easier.
• Apply exponent rules consistently: Use rules like the power of a power rule to simplify expressions step by step, eliminating extra work.
• Break down even further if needed: Sometimes, it helps to split a problem into smaller tasks, like taking intermediate roots or powers before final calculations.

Following these guidelines makes handling exponential expressions much easier and keeps calculations error-free.