Problem 77 - Algebra and Trigonometry

Question

perform the indicated operations. Simplify the result, if possible. $$ \frac{y^{-1}-(y+5)^{-1}}{5} $$

Step-by-Step Solution

Answer

The simplified form of the given equation is \(\frac{5-y}{5y(y+5)}\).

Step 1: Rewrite the negative exponents

Rewrite the expression replacing the negative exponent with its reciprocal form:\n\n\[ \frac{\frac{1}{y}-\frac{1}{y+5}}{5} \]

Step 2: Find common denominator

For the numerator, find a common denominator. To do this, multiple the two denominators together:\n\n\[ \frac{\frac{1(y+5)-1(y)}{y(y+5)}}{5} \]

Step 3: Simplify the numerator

Now simplify the numerator, expand where necessary and combine like terms:\n\n\[ \frac{\frac{5-y}{y(y+5)}}{5} \]

Step 4: Division of fractions

Interpret division as a multiplication (flip the fraction ''after'' the division sign and change the division to multiplication):\n\n\[ \frac{5-y}{y(y+5)}*\frac{1}{5} \]

Step 5: Final Simplification

Simplify the expression if possible by reducing fractions:\n\n\[ \frac{5-y}{5y(y+5)} \]

Key Concepts

Negative Exponents

Negative exponents can often seem tricky, but they really just involve a concept known as the reciprocal.
When you see a negative exponent, it tells you to take the reciprocal of the base. Simply put, when we have a term like \( y^{-1} \), it transforms into \( \frac{1}{y} \), turning the expression upside-down to change that negative sign.
So, whenever you see a negative exponent, remember it's all about flipping the base under a division sign, swapping signs for more friendly numbers.

Common Denominator

Finding a common denominator is essential when working with fractions because it allows us to combine them.
If you're adding or subtracting fractions, they must have the same denominator to proceed further.
Think of a common denominator as a shared stage that different fractions stand on, to perform together.

• In our exercise, to subtract \( \frac{1}{y} - \frac{1}{y+5} \), we need to multiply the denominators \( y(y+5) \) to find a shared ground.
• Then, adjust the numerators to match this new stage.
• This step brings the fractions together in harmony for easy subtraction.

Don't worry if this step takes some time—finding common denominators is a powerful tool that will streamline your fraction operations beautifully.

Dividing Fractions

Dividing fractions involves a simple trick: flip and multiply.
Instead of dividing by a fraction, you multiply by its reciprocal. This method leverages multiplication's simplicity over division's complexity.

• For instance, in our example, we changed \( \frac{5-y}{y(y+5)}/5 \) into \( \frac{5-y}{y(y+5)} \times \frac{1}{5} \).
• The division bar gives way to a multiplication sign, and we flip \( 5 \) into \( \frac{1}{5} \).

Remembering the flip-and-multiply strategy will make dividing fractions as easy as pie—each step more predictable and straightforward.

Simplifying Expressions

Simplification is like tidying up a room—it makes everything clearer and more manageable.
After performing operations like addition, subtraction, or division, your expression may have chances to be reduced or simplified further.
To simplify a fraction expression:

• Look for common factors in the numerator and denominator.
• Cancel them out wherever possible.
• Our exercise ends up with \( \frac{5-y}{5y(y+5)} \), well-organized into its simplest form.

Simplifying not only makes calculations easier but also helps ensure accuracy in math problems. So next time your math room feels cluttered, take a cleansing breath and simplify away!