Problem 27
Question
Use the product rule for exponents to simplify each expression. Write the results using exponents. $$ 5^{3} \cdot 5^{4} $$
Step-by-Step Solution
Verified Answer
The expression simplifies to \(5^7\).
1Step 1: Identify the Base and Exponents
Identify the base and the exponents in the expression. Here, both terms have the base 5, with exponents 3 and 4 respectively.
2Step 2: Apply the Product Rule for Exponents
The product rule for exponents states that when multiplying like bases, you add the exponents: \[ a^m \cdot a^n = a^{m+n} \].For \(5^3 \cdot 5^4\), this means:\[ 5^{3+4} \].
3Step 3: Simplify the Expression
Add the exponents 3 and 4 together to simplify:\[ 5^{3+4} = 5^7 \].
Key Concepts
Product RuleSimplifying ExpressionsAlgebraic Expressions
Product Rule
The product rule is a fundamental principle in algebra that comes into play when you are multiplying expressions with the same base. It's a nifty shortcut that helps us simplify multiplication by dealing directly with the exponents. Instead of multiplying the numbers outright, you simply add their exponents.
This rule works because exponents signify repeated multiplication. If you have \( a^m \) and \( a^n \), you're multiplying \( m \) factors of \( a \) with \( n \) factors of \( a \). The product rule simplifies this to \( a^{m+n} \).
For example, in the expression \( 5^3 \cdot 5^4 \), you don't multiply eighteen \(5's\). Using the product rule, simply add \(3\) and \(4\) to get \(5^7\). This saves time and keeps your work tidy.
This rule works because exponents signify repeated multiplication. If you have \( a^m \) and \( a^n \), you're multiplying \( m \) factors of \( a \) with \( n \) factors of \( a \). The product rule simplifies this to \( a^{m+n} \).
- Useful for quick calculations.
- Ensures that expressions are presented consistently.
- A great tool for minimizing steps in algebraic manipulation.
For example, in the expression \( 5^3 \cdot 5^4 \), you don't multiply eighteen \(5's\). Using the product rule, simply add \(3\) and \(4\) to get \(5^7\). This saves time and keeps your work tidy.
Simplifying Expressions
Simplifying expressions is a key part of solving algebraic problems. It involves reducing complex expressions into their simplest form, making them easier to work with. This can involve several different techniques, such as using rules for exponents or simplifying terms.
In our case, simplifying involves the product rule. By using this rule, you can quickly combine or simplify exponential terms with the same base.
Let's consider \( 5^3 \cdot 5^4 \). Without simplifying, you would multiply each factor of \(5\). But by adding the exponents, you reduce the work to just calculating \(5^7\). Always aim for simplicity, as it often leads to quicker and correct results.
In our case, simplifying involves the product rule. By using this rule, you can quickly combine or simplify exponential terms with the same base.
- Helps avoid errors in calculation.
- Makes evaluating expressions quicker.
- Reduces expressions to their simplest forms for clearer understanding.
Let's consider \( 5^3 \cdot 5^4 \). Without simplifying, you would multiply each factor of \(5\). But by adding the exponents, you reduce the work to just calculating \(5^7\). Always aim for simplicity, as it often leads to quicker and correct results.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. They're the language of algebra and can represent real-world situations or abstract concepts. Understanding how to manipulate these expressions is vital for solving equations and real-world problems.
Exponents add another layer to algebraic expressions, indicating repeated multiplication. When dealing with these, rules such as the product rule are incredibly useful.
An algebraic expression like \( 5^3 \cdot x^4 \) represents different operations on numbers and variables. When working with exponential expressions, knowing when and how to apply rules like the product rule ensures you can simplify effectively, making your work cleaner and your solutions clearer.
Exponents add another layer to algebraic expressions, indicating repeated multiplication. When dealing with these, rules such as the product rule are incredibly useful.
- Features constants, variables, and operations.
- Essential for expressing mathematical relationships.
- Can be solved, expanded, or simplified.
An algebraic expression like \( 5^3 \cdot x^4 \) represents different operations on numbers and variables. When working with exponential expressions, knowing when and how to apply rules like the product rule ensures you can simplify effectively, making your work cleaner and your solutions clearer.
Other exercises in this chapter
Problem 27
Find each product. See Example 2. $$ (2 p+7)(2 p-7) $$
View solution Problem 27
Multiply. See Example 2. $$ -4 t\left(t^{2}-7\right) $$
View solution Problem 27
Classify each polynomial as a monomial, a binomial, a trinomial, or none of these. See Example \(1 .\) $$ 6 x^{5}-x^{4}-3 x^{3}+7 $$
View solution Problem 27
Simplify each polynomial and write it in descending powers of one variable. $$ 4 x^{2} y+5-6 x^{3} y-3 x^{2} y+2 x^{3} y $$
View solution