Problem 48 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \frac{1}{y^{-7}} $$

Step-by-Step Solution

Answer

\(\frac{1}{y^{-7}}\) simplifies to \(y^7\).

Step 1: Understand the Problem

We need to simplify the expression \(\frac{1}{y^{-7}}\) while ensuring that there are no negative exponents involved. Negative exponents can be rewritten as positive exponents by adjusting their position in a fraction.

Step 2: Rewrite with Positive Exponents

To eliminate the negative exponent, we use the rule that \(x^{-a} = \frac{1}{x^a}\). Thus, \(y^{-7}\) is equal to \(\frac{1}{y^7}\). Therefore, \(\frac{1}{y^{-7}}\) becomes \(y^7\).

Step 3: Simplify Expression Further

Since we have rewritten \(\frac{1}{y^{-7}}\) as \(y^7\), we have fully simplified the expression. This expression is now in its simplest form with only positive exponents.

Key Concepts

Negative Exponents

Negative exponents might seem a bit intimidating at first, but they're quite simple when you break them down. When a number or a variable has a negative exponent, it's essentially telling us to "flip" it over in a fraction.
This is because

• the expression with a negative exponent (\(x^{-a}\)) can be rewritten as \(\frac{1}{x^a}\).

Think of it like this: the negative sign in the exponent is prompting us to take the reciprocal.
For example, \(y^{-3}\) isn't just a negative power of \(y\); it's actually \(\frac{1}{y^3}\). Remember that this rule helps us transform the expression, making the negative exponent vanish by shifting the base's position.

Positive Exponents

Once you've gotten rid of negative exponents by flipping their base in a fraction, what you're left with are positive exponents. Positive exponents simply mean you're multiplying the base by itself several times.
For example, \(y^5\) tells us to multiply \(y\) by itself five times.
Using positive exponents makes expressions easier to work with, especially when performing operations like multiplication or division.

• They are straightforward and more manageable when simplifying or grouping terms together.
• Maintaining positive exponents also aligns with conventional notation, keeping mathematical expressions clean and concise.

By converting negative exponents into positive ones, calculations and comparisons are simplified, which is why we often make this conversion in algebra.

Simplifying Expressions

After converting all exponents to positive, the next step is making the expression as simple as it can be. This involves reducing any redundant components or terms in the given equation.
The goal is to make your life easier while still retaining all the critical information in a tidy package.
So, how do we go about simplification?

• First, ensure all negative exponents are converted, as we discussed.
• Next, combine like terms and eliminate any unnecessary components.
• Mathematical expressions can involve distributing terms, factoring out common elements, or maybe even just plain old arithmetic.

By keeping your expressions simplified, you streamline your problem-solving process and reduce the risk of making errors in complex calculations. When your math is as simple as it can be, it's also at its most powerful.