Problem 50 - Elementary and Intermediate Algebra

Question

Use the product and quotient rules for exponents to simplify each expression. $$ \frac{h^{3} h^{6} h}{h^{9}} $$

Step-by-Step Solution

Answer

The simplified expression is \( h \).

Step 1: Apply the Product Rule

The product rule for exponents states that when you multiply two expressions with the same base, you add their exponents. Apply this rule to the terms in the numerator: \( h^{3} \times h^{6} \times h \). Add the exponents: \[ h^{3+6+1} = h^{10} \].Now, the expression simplifies to \[ \frac{h^{10}}{h^{9}} \].

Step 2: Apply the Quotient Rule

The quotient rule for exponents states that when you divide two expressions with the same base, you subtract the exponents. Apply the rule to the simplified expression:\( \frac{h^{10}}{h^{9}} \).Subtract the exponents: \[ h^{10-9} = h^{1} \], which further simplifies to \( h \).

Key Concepts

Product Rule

The product rule is an essential part of simplifying expressions with exponents. It's a simple yet powerful tool. When multiplying two or more terms that have the same base, you can simply add their exponents.
This rule can greatly simplify your calculations. Imagine you have terms like \( h^3 \times h^6 \times h \). Since all terms have the same base \( h \), apply the product rule: add up the exponents \(3\), \(6\), and \(1\), giving you \( h^{3+6+1} = h^{10} \).
Here's a quick breakdown of the product rule steps:

• Check if bases are the same.
• Add the exponents together.
• Simplify to a single term with the added exponent.

By using the product rule, you can transform complicated multiplications into straightforward expressions, making further calculations easier.

Quotient Rule

The quotient rule for exponents is equally useful when simplifying expressions involving division. When you divide terms with the same base, subtract the exponent of the denominator from the exponent of the numerator. This will give you a quick simplified expression.
For example, take the expression \( \frac{h^{10}}{h^{9}} \). Here, both terms have the same base \( h \), so you subtract the exponents: \( 10 - 9 = 1 \), leaving you with \( h^1 \), or simply \( h \).
Steps for applying the quotient rule:

• Ensure both terms have the same base.
• Subtract the exponents (numerator exponent minus denominator exponent).
• Write the result as a single term with the subtracted exponent.

Using the quotient rule efficiently reduces complex fractional expressions to simple terms, which are easier to comprehend and work with.

Simplifying Expressions

Simplifying expressions with exponents involves using rules to make the expression as straightforward as possible. The goal is to combine like terms and apply rules such as the product and quotient rules we discussed earlier.
When starting with an expression like \( \frac{h^3 h^6 h}{h^9} \), the idea is to first simplify the numerator using the product rule \( h^3 \times h^6 \times h = h^{10} \). Then, apply the quotient rule to divide by the denominator \( h^9 \), giving \( h^{10-9} = h^1 \), which is simply \( h \).
In essence, simplifying expressions means:

• Applying the product rule to multiply terms effectively.
• Using the quotient rule to divide terms correctly.
• Combining like terms to achieve the simplest form of the expression.

Simplifying is about making expressions easier to handle and less prone to errors, which is crucial in solving mathematical problems efficiently.