Problem 33 - Algebra 2

Question

Simplify each number. $$(-32)^{\frac{6}{5}}$$

Step-by-Step Solution

Answer

The simplified form of \( (-32)^{\frac{6}{5}} \) is \(64\).

Step 1: Identify the Base and the Exponent

First identify the base and the exponent in \( (-32)^{\frac{6}{5}} \). The base is \(-32\) and the exponent is \(\frac{6}{5}\). Note that the base is a negative number and the exponent is a fraction.

Step 2: Simplify the Fraction

\(-32\) raised to an exponent of \(\frac{6}{5}\) can be rewritten as \( ( -32^{\frac {1}{5} } ) ^6 \). Now simply calculate the 5th root of \(-32\) which results in \(-2\). So the expression can be simplified to \( (-2)^6 \).

Step 3: Raise to the power

Now, raise \(-2\) to the power 6, which results in \(64\). The reason why the result is positive despite the base being negative is because an even power makes a negative number positive.

Key Concepts

Negative Bases

When dealing with **negative bases**, especially in mathematics involving exponents, it's important to remember how they behave differently than positive bases. A negative base means that the value itself is carrying a negative sign. In the context of exponents, you need to consider how the sign impacts the result:
- If the exponent is an **even number**, the result is positive. This is because multiplying a negative number by itself an even number of times cancels out the negative sign. For example, \((-2)^6 = 64\) because the six negative numbers multiply to make a positive number.
- If the exponent is an **odd number**, the result remains negative. For example, \((-3)^3 = -27\) as multiplying a negative number three times will still yield a negative result.

A key takeaway is to always check whether your exponent is even or odd when working with negative bases, as it determines the sign of your final answer.

Fractional Exponents

With **fractional exponents** we are essentially doing two operations: roots and powers. Fractional exponents are a way of expressing both a root and an exponent in one term. The numerator of the fraction is the power, while the denominator is the root to be taken. For instance, in the expression \((-32)^{\frac{6}{5}}\), it can be broken down as follows:
- The denominator \(5\) indicates that you should take the **5th root** of the base, \(-32\).
- The numerator \(6\) indicates that you raise the result of the 5th root to the **6th power**.

This simplification process is handy because it allows breaking down a relatively complex problem into more manageable steps. By computing roots first, calculations become simpler and easier to handle, especially for negative bases.

Exponent Simplification

**Exponent simplification** is a powerful skill that helps make complex equations easier to manipulate and solve. The process of simplifying exponents involves breaking down expressions into their simplest form by applying known mathematical rules. For example, simplifying \((-32)^{\frac{6}{5}}\) involves turning it into a series of simpler computations:
1. Rewrite \((-32)^{\frac{6}{5}}\) as \(( (-32)^{\frac{1}{5}} )^{6}\). This separates the root and the power into two steps.
2. Compute the **5th root** of \(-32\) to obtain \(-2\).
3. Finally, raise \(-2\) to the **6th power**. Since the exponent is even, the result is positive, yielding \64\.

The process of exponent simplification not only helps in understanding how the expression can be rewritten to achieve an answer but also demonstrates the relationships between roots and powers, ultimately aiding in mastering the use of exponents in algebra.