Problem 55 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{y^{9}}{y^{4}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{y^9}{y^4}\) using the product rule and quotient rule of exponents. Answer: $$y^5$$

Step 1: Identify the base and exponents in the given expression

Take a look at the given expression: \(\frac{y^9}{y^4}\). Here, the base is 'y' and, in the numerator, the exponent is '9'. In the denominator, the exponent is '4'.

Step 2: Apply the quotient rule of exponents

The quotient rule of exponents states that when we divide two expressions with the same base, we can subtract their exponents. So in our given expression, \(\frac{y^9}{y^4}\), we subtract the exponent of the denominator from the exponent of the numerator: \(9 - 4 = 5\).

Step 3: Rewrite the simplified expression

Now, write the simplified expression with the base 'y' and the new exponent '5': $$y^5$$ So, the simplified expression is: $$y^5$$.

Key Concepts

Simplifying Exponents

When working with algebra, you'll often encounter expressions with exponents that need to be simplified. Simplifying exponents means reducing expressions to their most basic form without changing their values. Understanding the properties of exponents is crucial in performing these simplifications correctly. Let's consider an example.

The expression \(\frac{y^9}{y^4}\) can be simplified using the quotient rule of exponents. This rule is a shortcut that helps us divide expressions with the same base. In this case, both the numerator and denominator have the base \(y\). According to the quotient rule, we subtract the exponent in the denominator from the exponent in the numerator, resulting in \(y^{9-4} = y^5\). So, the expression \(\frac{y^9}{y^4}\) simplifies to \(y^5\), which is much easier to work with in subsequent calculations.

When simplifying exponents, always remember to look for opportunities to apply the exponent rules, such as the quotient rule, to make the expression as simple as possible.

Exponent Rules

Understanding Exponent Rules

Exponent rules, also known as the laws of exponents, are pivotal in simplifying algebraic expressions involving powers. These rule sets include the product rule, quotient rule, power of a power rule, and others. For example, the product rule tells us that when multiplying two expressions with the same base, we add their exponents. In contrast, the quotient rule, which was applied in our exercise, instructs us to subtract exponents when dividing.

Let's highlight the exponent rules:

• Product Rule: \( x^a \cdot x^b = x^{a+b} \)
• Quotient Rule: \( \frac{x^a}{x^b} = x^{a-b}\)
• Power of a Power Rule: \( (x^a)^b = x^{a\cdot b}\)
• Zero Exponent Rule: \( x^0 = 1\), for any non-zero \(x\)
• Negative Exponent Rule: \( x^{-a} = \frac{1}{x^a}\), given \(x\) is not zero

By mastering these rules, you gain the ability to navigate through complex problems and simplify them down to manageable expressions. Always apply the appropriate rule based on the operation you are performing and the form of the expression you are dealing with.

Algebraic Expressions

Navigating Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. For instance, \(\frac{y^9}{y^4}\) is an expression composed of a single variable \(y\) and exponents which are whole numbers. Simplifying algebraic expressions is a fundamental skill in algebra.

As you become more comfortable with using exponent rules, you will find that algebraic expressions become more manageable. A strong understanding of variables and the ability to recognize patterns is essential. Remember, the goal when simplifying an expression is to make it as straightforward as possible, which often means performing operations to eliminate exponents or to reduce the number of terms in the expression.

Also, always check your final answer to make sure all possible simplifications have been made. This includes looking for common factors that can be divided out and combining like terms. The simplicity and elegance of your final answer often directly correlate with the depth of your understanding of the rules and concepts of algebra.