Problem 34 - Algebra A Combined Function

Question

Write with positive exponents. Simplify if possible. $$ (-16)^{-5 / 4} $$

Step-by-Step Solution

Answer

The expression with positive exponents and in simplified form is \(\frac{1}{32 \cdot e^{5i\pi/4}}\).

Step 1: Understanding the Problem

We need to write the expression \((-16)^{-5/4}\) with positive exponents. The problem also involves taking roots because of the fractional exponent.

Step 2: Using the Property of Negative Exponents

A negative exponent \(a^{-b}\) can be rewritten as \(\frac{1}{a^b}\). Therefore, \((-16)^{-5/4}\) is equivalent to \(\frac{1}{(-16)^{5/4}}\).

Step 3: Rewriting the Fractional Exponent

Remember that \(a^{m/n}\) can be interpreted as \(\sqrt[n]{a^m}\). For \((-16)^{5/4}\), this means we calculate the fourth root of \((-16)^5\).

Step 4: Dealing with the Negative Base and Fractional Exponent

The fourth root of a negative number is not real because no real number raised to the fourth power can give a negative result. Therefore, we extend our calculation to complex numbers.

Step 5: Simplifying in the Complex Domain

With complex numbers, we rewrite \((-16)\) as \((16e^{i\pi})\) following Euler's formula where the fourth root of \(16e^{i\pi}\) will become \((2\cdot e^{i\pi/4})\). Then we raise this result to the 5th power: \([2 \cdot e^{i\pi/4}]^5 = 32 \cdot e^{5i\pi/4}\).

Step 6: Final Expression

Therefore, the expression \((-16)^{-5/4}\) simplifies to \(\frac{1}{32 \cdot e^{5i\pi/4}}\), which is the `complex` number representation of the original expression with positive exponents.

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but they are all about turning the number upside down. When you see something like \(a^{-b}\), it simply means you flip it over to become \(\frac{1}{a^b}\). This process is called taking the reciprocal. For instance, \((-16)^{-5/4}\) becomes \(\frac{1}{(-16)^{5/4}}\) when the negative exponent is addressed.

• Step one: Identify the negative exponent.
• Step two: Flip it to make it positive. For example, changing negative exponents into a fraction can make calculations much more manageable.

Remember, handling negative exponents is like turning something that is a small fraction into something potentially larger (and vice versa). Keep practicing, and it will become second nature!

Fractional Exponents

Fractional exponents combine the power and roots in a neat and compact form. When you see an expression with fractional exponents like \((-16)^{5/4}\), remember it involves both raising the number to a power and taking the root.

• Think of \(a^{m/n}\) as \(\sqrt[n]{a^m}\). This tells you to first raise \(a\) by \(m\) and then take the \(nth\) root.
• For \((-16)^{5/4}\), you'd compute \(\sqrt[4]{(-16)^5}\). First, raise \(-16\) to the 5th power, and then take the fourth root.

This method can simplify complex calculations, reducing them into easier steps of powers and roots. So, fractional exponents allow us to express and work with both exponents and roots seamlessly in a single mathematical operation.

Roots of Negative Numbers

When working with roots of negative numbers, things can get interesting. Real numbers do not include the even roots of negative numbers because no real number squared, for example, can produce a negative number. Thus, we extend our number system to complex numbers.

• For example, the equation \(x^2 = -1\) has no real solution, but in the complex numbers, it has the solution \(i\), where \(i^2 = -1\).
• Taking roots, like the fourth root of a negative number (e.g., \(-16\)), involves moving to the complex plane. There, you can use concepts like Euler’s formula to simplify expressions.

In summary, while negative numbers present challenges for even roots in real numbers, complex numbers offer solutions, giving equations like \((-16)^{1/4}\) meaning and a path to an answer.

Euler's Formula

Euler's formula \(e^{ix} = \cos(x) + i\sin(x)\) is a vital tool in bridging trigonometry and complex numbers. It helps in simplifying complex expressions involving exponential functions.

• For example: Rewrite \((-16)\) as \(16e^{i\pi}\). This transformation is based on the fact that any complex number can be represented with Euler's formula.
• Applying Euler's formula allows expression simplifications and makes taking roots of negative numbers in the complex domain possible.
• In the problem, using Euler’s formula reveals \((-16)\) as a complex number, and calculating powers becomes more accessible through polar and exponential forms.

Euler's formula is a powerful tool and a key concept in solving problems involving roots and powers of negative numbers in the complex plane. Practice using this tool will improve your skill in handling complex number problems.