Problem 38 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 16^{-3 / 4} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{8} \).

Step 1: Rewrite the Expression as a Fraction Exponent

The given expression is \( 16^{-3/4} \). By the properties of exponents, we can interpret the negative exponent as the reciprocal: \( 16^{-3/4} = \frac{1}{16^{3/4}} \). This provides a more manageable expression to evaluate.

Step 2: Simplify the Base Using Roots

Next, simplify \( 16^{3/4} \) as \( (16^{1/4})^3 \). Since \( 16^{1/4} \) represents the fourth root of 16, we find \( 16^{1/4} = 2 \) because \( 2^4 = 16 \). Thus, \( 16^{3/4} = (2)^3 \).

Step 3: Calculate the Result

Now substitute back: \( (2)^3 = 8 \), so \( 16^{3/4} = 8 \). Thus, \( \frac{1}{16^{3/4}} = \frac{1}{8} \).

Step 4: Conclusion: Express the Final Answer

The value of the expression \( 16^{-3/4} \) is \( \frac{1}{8} \).

Key Concepts

Negative Exponents

In mathematics, a negative exponent indicates a reciprocal. When we see an exponent like \( a^{-n} \), it means we take the reciprocal of \( a^{n} \). For example, the expression \( 16^{-3/4} \) transforms to \( \frac{1}{16^{3/4}} \). This reversal simplifies dealing with expressions that have negative powers.
An intuitive way to understand this is to think of a negative sign in an exponent as a 'flip' operation. Rather than multiplying the base by itself a negative number of times (which doesn’t make direct sense), we invert the whole process by taking the reciprocal. This interpretation helps simplify calculations and makes them more understandable.
Remember, the basic rule is:
- \( a^{-n} = \frac{1}{a^{n}} \).
Apply this whenever you see a negative exponent, to start simplifying the problem.

Root Simplification

Root simplification is a useful technique to simplify expressions with fractional exponents. For example, \( 16^{3/4} \) involves two operations: the root and exponentiation. The fraction \( \frac{3}{4} \) signifies both a root (the bottom number) and a power (the top number).
What we do first is identify the root. The denominator of the exponent, \( 4 \), tells us to take the fourth root of \( 16 \). Since \( 16 = 2^4 \), the fourth root of \( 16 \) is \( 2 \). This gives us \( 16^{1/4} = 2 \). Now, we use the power part: raise \( 2 \) to the third power (the numerator), resulting in \( 2^3 = 8 \). Hence, \( 16^{3/4} = 8 \).
This method shows how useful roots are for simplifying expressions with fractional exponents, turning them into whole numbers.

Exponent Properties

Exponent properties form the foundation for manipulating and simplifying expressions involving powers. They help in breaking down complex problems into manageable steps.
Some key properties include:

• The product of powers: \( a^m \cdot a^n = a^{m+n} \), which combines like bases by adding their exponents.
• The power of a power: \( (a^m)^n = a^{m\cdot n} \), which represents applying two exponents sequentially by multiplying them.
• The power of a product: \( (ab)^n = a^n \cdot b^n \), which distributes an exponent across a product.
• The reciprocal, helpful for negative exponents: \( a^{-n} = \frac{1}{a^n} \), which we’ve seen represents taking reciprocals.

These rules simplify expressions and provide a clear path through mathematical problems. Knowing these properties is essential when dealing with both whole and fractional exponents, helping you solve problems like \( 16^{-3/4} \) step by step.