Problem 16 - Elementary Algebra

Question

When simplifying the terms for the following problems, write each so that only positive exponents appear. $$ \frac{(y+1)^{3}(y-3)^{4}}{(y+1)^{5}(y-3)^{-8}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(\frac{(y-3)^{12}}{(y+1)^2}\).

Step 1: Eliminate the negative exponent

We will use the property \(a^{-m} = \frac{1}{a^m}\) to eliminate the negative exponent in \((y-3)^{-8}\): $$ \frac{(y+1)^{3}(y-3)^{4}}{(y+1)^{5}(y-3)^{-8}} = \frac{(y+1)^{3}(y-3)^{4}}{(y+1)^{5}} \cdot \frac{1}{(y-3)^{-8}} $$ Now, applying the property to \((y-3)^{-8}\), we get: $$ \frac{(y+1)^{3}(y-3)^{4}}{(y+1)^{5}} \cdot \frac{1}{\frac{1}{(y-3)^{8}}} = \frac{(y+1)^{3}(y-3)^{4}}{(y+1)^{5}} \cdot (y-3)^{8} $$

Step 2: Simplify the remaining exponents

Now we have eliminated the negative exponents. Next, we will simplify the remaining terms using the properties of exponents: $$ \frac{(y+1)^{3}(y-3)^{4}}{(y+1)^{5}} \cdot (y-3)^{8} = (y+1)^{3-5}(y-3)^{4+8} $$ Applying the arithmetic for the exponents, we get: $$ (y+1)^{-2}(y-3)^{12} $$

Step 3: Eliminate the negative exponent again

We still have a negative exponent in the expression \((y+1)^{-2}\), which needs to be eliminated using the property \(a^{-m} = \frac{1}{a^m}\): $$ (y+1)^{-2}(y-3)^{12} = \frac{(y-3)^{12}}{(y+1)^2} $$

Step 4: Write the final simplified form

Now all the exponents in our expression are positive. So the simplified form is: $$ \frac{(y-3)^{12}}{(y+1)^2} $$

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions involves rewriting them in their simplest form. This usually means performing all possible arithmetic operations and algebraic manipulations to make the expression shorter and easier to understand or use. In expressions involving exponents, one important aim is to have all exponents as positive when asked to simplify. Negative exponents can make expressions look more complicated, so we use specific properties of exponents to simplify them.

To simplify an expression efficiently, follow these steps:

• Resolve negative exponents first, as they tend to add complexity to the expression.
• Combine like terms, which might involve adding or subtracting exponents when terms are multiplied or divided.
• Convert any remaining negative exponents into positive ones to make the final expression look cleaner and less bulky.

In the example solved, we take an expression with a mix of positive and negative exponents and gradually resolve them using exponential rules until only positive exponents are left.

Properties of Exponents

The properties of exponents make it easier to manipulate expressions containing powers. These allow us to simplify expressions further and make calculations more manageable. Here are some key properties we used in the solution:

• Product of Powers: This property states that when you multiply two expressions with the same base, you can add their exponents. For instance, \((a^m) \times (a^n) = a^{m+n}\).
• Quotient of Powers: This property involves division with the same base, allowing you to subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: When raising a power to another power, you multiply the exponents, i.e., \((a^m)^n = a^{m \cdot n}\).
• Negative Exponents: Any base raised to a negative exponent is equivalent to one over the base raised to the positive exponent, such as \(a^{-m} = \frac{1}{a^m}\).

By mastering these properties, you can handle expressions involving exponents more confidently and efficiently.

Negative Exponents

Negative exponents often seem tricky, but once you understand the principle behind them, they become much easier to work with. A negative exponent indicates that the base should be taken as the reciprocal raised to the corresponding positive exponent.

Here's how to tackle negative exponents:

• A negative exponent can be rewritten as a fraction, \(a^{-m} = \frac{1}{a^m}\). For example, \(x^{-2} = \frac{1}{x^2}\).
• When dealing with fractions, a negative exponent in the numerator means you can move the base to the denominator with a positive exponent.
• Conversely, a negative exponent in the denominator can be brought up to the numerator as a positive exponent.

This concept is pivotal in simplifying expressions so that all terms present only positive exponents. In our solution, moving terms with negative exponents between the numerator and denominator helped us achieve a neater expression free from negative exponents.