Problem 31 - Beginning and Intermediate Algebra

Question

Simplify. $$\frac{y^{35}}{y^{31}}$$

Step-by-Step Solution

Answer

The simplified fraction is \(y^4\).

Step 1: Identify the rule to apply

We need to apply the quotient rule to simplify the given fraction. The quotient rule states that \( a^{m} / a^{n} = a^{m-n} \), where a is the base, m and n are the exponents.

Step 2: Apply the quotient rule

In our problem, the base is y, the exponent in the numerator is 35, and the exponent in the denominator is 31. Using the quotient rule, we need to subtract the exponents: \[ \frac{y^{35}}{y^{31}} = y^{35 - 31} \]

Step 3: Calculate the result

Now, subtract the exponents: \[ y^{35 - 31} = y^4 \] The simplified fraction is \(y^4\).

Key Concepts

Quotient Rule

The quotient rule is a useful tool in mathematics for simplifying division problems that involve exponents. It states that when you have the same base raised to two different powers, you can simplify by subtracting the exponents of the base.
For example, in the expression \( \frac{y^{35}}{y^{31}} \), both the numerator and the denominator have the base \( y \).
The quotient rule helps simplify this expression by subtracting the exponent in the denominator (31) from the exponent in the numerator (35).
This results in \( y^{35 - 31} = y^4 \).

So, generally speaking, the quotient rule is expressed as:
\[ \frac{a^m}{a^n} = a^{m-n} \]
It's important to remember that this rule only applies when the bases are the same. This makes it an incredibly useful shortcut in algebra, saving you from the tedious task of writing out all the factors.

Simplifying Expressions

Simplifying expressions in mathematics makes them easier to understand and work with. It involves reducing them to their simplest form without changing their value.
In our example, the expression \( \frac{y^{35}}{y^{31}} \) seems complex initially but simplifies to a single power of \( y \) when we apply exponent rules.

Simplifying involves

• Removing or combining like terms
• Using exponent rules like the quotient rule in fractions

By performing these actions, the expression becomes simpler and more straightforward.
Consider that by simplifying \( y^{35 - 31} \), we get \( y^4 \), which is a much cleaner and easier form to work with.

Simplifying expressions not only helps in solving equations accurately but also makes it more manageable and less error-prone.
Always ask yourself, "Can this expression be made simpler?" It usually can be, and when it is, working with it becomes much easier.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions.
These can involve numbers, variables, and exponents, making them an important part of algebra. They help in expressing complex relationships in a compact form.

Just like numerical fractions, algebraic fractions can also be simplified.

• Use the rules of exponents (like the quotient rule)
• Cancel out common factors in both the numerator and denominator

In our original exercise \( \frac{y^{35}}{y^{31}} \), you start by using the exponent rule to simplify the fraction to \( y^4 \).
Simplifying algebraic fractions enables clearer and more efficient problem solving, especially as expressions become more complex.

Always simplify whenever possible, as it often leads to a more elegant and understandable solution. Remember, practice makes perfect with these types of problems, so keep practicing to master the art of simplifying algebraic fractions.