Question

Evaluate the expression without using a calculator. (See Example 2.) \(16^{-7 / 4}\)

Short answer

The evaluation of \(16^{-7 / 4}\) without using a calculator results in \(1 / 128\).

Step-by-step solution

Understand Exponent Rules

Let's first understand how to handle the exponent. A negative exponent indicates that the base is on the wrong side of the fraction line, so you can start by flipping the base to the other side. As for the fractional exponent, the numerator represents the power to which the number is to be raised, while the denominator represents the root.

Handle Negative Exponent

So in the given problem \(16^{-7 / 4}\), flip the base to deal with the negative exponent. This results in \(1 / 16^{7 / 4}\).

Handle Fractional Exponent

Next, the expression \(1 / 16^{7 / 4}\) can be rewritten using the rules of fractional exponents. The 7 in the numerator of the fractional exponent indicates that 16 has to be raised to the power of 7, and the 4 in the denominator tells us that the fourth root of 16 needs to be taken. So, considering that the fourth root of 16 is 2, this yields: \(1 / 2^{7}\).

Final Calculation

Finally, \(1 / 2^{7}\) is easy to calculate: it's \(1 / 128\).

Key concepts

Negative Exponents

Negative exponents can be a bit quirky but are quite important in mathematics. When you encounter a negative exponent, it means you have to "flip" the base. For example, if you have a base like 16 with an exponent of -1, that’s equivalent to saying \( \frac{1}{16^1} \).
In a broader sense, 16 raised to the power of -7/4, such as in this exercise, means you first flip the base and then process the fractional part of the exponent.
This flipping action stems from the rule that \( a^{-n} = \frac{1}{a^n} \), which is true for any positive number 'a' and any positive integer 'n'. Thus, negative exponents turn what could be a large number into a small "fractional" one by essentially taking the reciprocal of the base.

Exponent Rules

Exponent rules help us understand how powers behave under different circumstances. These rules are crucial for simplifying expressions and solving equations. Here are a few key ones:

• **Product of Powers Rule:** This rule tells us that when you’re multiplying two powers with the same base, you add the exponents. For instance, \(a^m \times a^n = a^{m+n}\).
• **Quotient of Powers Rule:** When dividing powers with the same base, subtract the exponents: \[\frac{a^m}{a^n} = a^{m-n}\].
• **Power of a Power Rule:** If you need to raise a power to another power, you multiply the exponents, such as \( (a^m)^n = a^{m\cdot n} \).
• **Fractional Exponent Rule:** This tells us that a fractional exponent, \( a^{\frac{m}{n}} \), is equivalent to taking the n-th root of 'a' to the m-th power.

Applying these rules helps simplify the calculations and make sense of expressions with complicated components.

Roots and Powers

Roots and powers are two sides of the same mathematical coin. While powers tell us how many times to multiply a number by itself, roots help us find a number which when multiplied by itself a certain number of times gives us the original number.
For example, the square root of 9 is 3 because 3 squared, or \( 3^2 \), equals 9. Similarly, fractional exponents like \( a^{\frac{1}{n}} \) represent the n-th root of 'a'.
This is particularly useful when dealing with fractional exponents in exercises. So for an expression like \( 16^{\frac{7}{4}} \,\) you can first find the 4th root of 16, which is 2, and then raise it to the power of 7. Remembering the interplay between roots and powers provides a robust method for solving more complex problems.