Problem 88 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \left(\frac{1}{x^{8}}\right)^{-1 / 4} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{x^2}\).

Step 1: Apply the Negative Exponent Rule

The expression has a negative exponent, so we can flip the base to eliminate the negative sign. Using the rule \(a^{-n} = \frac{1}{a^n}\), we find: \(\left(\frac{1}{x^8}\right)^{-1/4} = x^8\).

Step 2: Apply the Power Rule for Exponents

Next, apply the power rule for exponents which states that \((a^m)^n = a^{m \times n}\). Rewrite the expression with a rational exponent: \((x^8)^{-1/4} = x^{8 \times (-1/4)} = x^{-2}\).

Step 3: Simplify to a Positive Exponent

Since we have another negative exponent, apply the rule \(a^{-n} = \frac{1}{a^n}\) to simplify: \(x^{-2} = \frac{1}{x^2} \).

Key Concepts

Negative Exponent Rule

Understanding negative exponents can be much easier with a simple rule: the negative exponent rule. When you see a negative exponent in an expression, it usually means that the term is a reciprocal. In math terms, for any non-zero number \(a\) and a positive integer \(n\), the rule is:

• \(a^{-n} = \frac{1}{a^n}\)

To flip that around, if \(a\) is written as a fraction like \(\frac{1}{a^n}\), you can also express it as \(a^{-n}\). This rule helps eliminate negative signs by flipping the base to its reciprocal.
For our problem, we start with an expression that looks a bit complex: \(\left(\frac{1}{x^{8}}\right)^{-1 / 4}\). The first thing to notice is the negative exponent \(-\frac{1}{4}\). To apply the negative exponent rule, you flip the base to get rid of that negative sign entirely. The expression \(\frac{1}{x^8}\) becomes \(x^8\) when flipped, because you take the reciprocal.

Power Rule for Exponents

Next, we use the power rule for exponents to simplify the expression further. This rule makes working with powers and exponents straightforward. It says that when you raise a power to another power, you multiply the exponents.
For any non-zero number \(a\) and real numbers \(m\) and \(n\), the rule is:

• \((a^m)^n = a^{m \cdot n}\)

Applying this rule helps you break down complex exponent expressions with ease. In our exercise, once we flipped the expression to \(x^8\), we noticed it was raised to the power of \(-\frac{1}{4}\). So we multiply the exponents:

• \(8\times (-\frac{1}{4})= -2\)

Therefore, \((x^8)^{-1/4}\) simplifies to \(x^{-2}\). This is still not the simplest form, because of the negative exponent.

Simplifying Expressions

Finally, simplify the expression to ensure your answer is entirely positive. A negative exponent, although valid, is not often used in a final simplified form. So, the next step is crucial for converting it to a positive exponent. We already have \(x^{-2}\).
To make \(x^{-2}\) easier to work with, use the negative exponent rule again:

• \(x^{-2} = \frac{1}{x^2}\)

This step "flips" the expression around to eliminate the negative sign, giving us a final answer that is simple and easy to understand, with only positive exponents. This makes the expression much more intuitive and ready for any further operations or evaluations you may need to perform.