Problem 38 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{1}{8} \).

Step 1: Understanding the Negative Exponent

The expression involves a negative exponent, which means \[ 16^{-3/4} = \frac{1}{16^{3/4}} \]This step involves understanding that a negative exponent indicates taking the reciprocal of the positive exponent.

Step 2: Simplify the Exponent Form

Now, we focus on simplifying \( 16^{3/4} \). The exponent \( 3/4 \) refers to the cube of the fourth root of 16. We can rewrite it as:\[ 16^{3/4} = (16^{1/4})^3 \] where \( 16^{1/4} \) denotes the fourth root of 16.

Step 3: Calculate the Fourth Root

Calculate the fourth root of 16. Since 16 can be written as \( 2^4 \), we have:\[ 16^{1/4} = (2^4)^{1/4} = 2^{4/4} = 2 \]Thus, the fourth root of 16 is 2.

Step 4: Cube the Fourth Root

Now, we cube the result from the previous step:\[ (16^{1/4})^3 = 2^3 = 8 \] This gives us the value of \( 16^{3/4} \) as 8.

Step 5: Take the Reciprocal

Finally, use the result from Step 4 to calculate the reciprocal, which was derived in Step 1:\[ 16^{-3/4} = \frac{1}{8} \]This step concludes the calculations and provides the final answer to the expression.

Key Concepts

Negative Exponents

When we encounter a negative exponent, it often looks intimidating, but it's just a different way to express fractions. A negative exponent tells us to take the reciprocal of the base raised to the corresponding positive exponent. For example, when you see \( 16^{-3/4} \), it simply means \( \frac{1}{16^{3/4}} \).

• To deal with negative exponents, remember that the minus sign is there to tell you to "flip" the fraction.
• This "flipping" process is similar whether the exponent is a whole number or a rational number.

By converting negative exponents into reciprocals, they become easier to manage in calculations.

Rational Exponents

Rational exponents, such as \( 16^{3/4} \), are another way of expressing roots and powers. The fraction in the exponent \( \frac{3}{4} \) means the fourth root of a number raised to the third power. This can be expressed as \( (16^{1/4})^3 \).

• The denominator in the rational exponent indicates the root. In our example, 4 tells us we're taking the fourth root.
• The numerator indicates the power. Here, 3 means that the fourth root result is to be cubed.

Using rational exponents often simplifies operations, especially when dealing with roots and powers together.

Simplifying Exponents

Simplifying expressions with exponents transforms complex problems into simpler ones. Continuing with \( 16^{3/4} \), we see this simplification is broken down into manageable steps:

• First, you take the fourth root of 16, which means finding a number that, when multiplied by itself four times, gives 16. Knowing that \( 16 = 2^4 \), the fourth root is 2.
• Then, cube this result (that is, multiply it by itself twice more): \( 2^3 = 8 \).

This step-by-step simplification helps in understanding not only the problem at hand but also builds confidence in tackling other exponential expressions.

Roots and Powers

Understanding the relationship between roots and powers is key to mastering exponents.
Any number raised to a power is a repeated multiplication, while roots are the opposite: finding what number can be multiplied several times to yield the original number.
In the expression \( 16^{3/4} \):

• The fourth root: \( 16^{1/4} = 2 \) because \( 2^4 = 16 \).
• The power: \( (2)^3 = 8 \) which completes the calculation.

Visualizing roots as finding the "building blocks" of a number and powers as "constructing" using those blocks can make these concepts more intuitive.