Problem 100
Question
Simplify each expression. $$ \frac{x^{12} y^{13}}{x^{5} y^{7}} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \( x^{7} y^{6} \).
1Step 1: Identify the Components
The given expression is \( \frac{x^{12} y^{13}}{x^{5} y^{7}} \). It consists of two variables, \( x \) and \( y \), each raised to different powers in the numerator and denominator.
2Step 2: Apply the Quotient Rule for Exponents
According to the quotient rule for exponents, \( \frac{a^{m}}{a^{n}} = a^{m-n} \). We will apply this rule separately for \( x \) and \( y \).
3Step 3: Simplify the x Component
For the \( x \) variable, apply the quotient rule: \( \frac{x^{12}}{x^{5}} = x^{12-5} = x^{7} \).
4Step 4: Simplify the y Component
For the \( y \) variable, apply the quotient rule: \( \frac{y^{13}}{y^{7}} = y^{13-7} = y^{6} \).
5Step 5: Combine Simplified Components
Combine the simplified components to form the final expression: \( x^{7} y^{6} \).
Key Concepts
ExponentsQuotient Rule for ExponentsAlgebraic Expressions
Exponents
Exponents are a way to express how many times a number, known as the base, is multiplied by itself. If you see a number such as \( a^n \), it means you multiply the base \( a \) by itself \( n \) times. This can simplify expressions greatly and makes it easy to manage large numbers or complex algebraic expressions.
Consider an expression like \( x^{12} \), which tells you that \( x \) is multiplied by itself 12 times. Similarly, \( y^{13} \) indicates \( y \) is multiplied by itself 13 times. Understanding this concept lays the foundation for using and manipulating more complex expressions with exponents.
Some important points to remember about exponents:
Consider an expression like \( x^{12} \), which tells you that \( x \) is multiplied by itself 12 times. Similarly, \( y^{13} \) indicates \( y \) is multiplied by itself 13 times. Understanding this concept lays the foundation for using and manipulating more complex expressions with exponents.
Some important points to remember about exponents:
- When you multiply like bases, you add their exponents together, such as \( a^m \times a^n = a^{m+n} \).
- When you divide like bases, you subtract the exponent of the denominator from the exponent of the numerator.
- Any number raised to the power of zero is 1, except zero itself.
- Negative exponents represent the reciprocal of the base raised to the opposite positive exponent.
Quotient Rule for Exponents
The quotient rule for exponents is a very handy tool when simplifying expressions that involve division of the same base with different powers. The rule states that for any base \( a \), and exponents \( m \) and \( n \), the expression \( \frac{a^m}{a^n} = a^{m-n} \) holds true.
This means you can subtract the exponent in the denominator from the exponent in the numerator when the bases are the same, thereby simplifying the expression. It essentially shows how many times the base remains after considering both the numerator and the denominator.
In our example, for the expression \( \frac{x^{12}}{x^5} \), using the quotient rule gives us \( x^{12-5} = x^7 \). Similarly, for \( \frac{y^{13}}{y^7} \), applying the quotient rule results in \( y^{13-7} = y^6 \).
Always make sure that the bases are the same before applying this rule, as it enables you to simplify the expression safely.
This means you can subtract the exponent in the denominator from the exponent in the numerator when the bases are the same, thereby simplifying the expression. It essentially shows how many times the base remains after considering both the numerator and the denominator.
In our example, for the expression \( \frac{x^{12}}{x^5} \), using the quotient rule gives us \( x^{12-5} = x^7 \). Similarly, for \( \frac{y^{13}}{y^7} \), applying the quotient rule results in \( y^{13-7} = y^6 \).
Always make sure that the bases are the same before applying this rule, as it enables you to simplify the expression safely.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They can range from very simple, containing only one variable and one operation, to very complex, involving multiple variables and several operations.
Simplifying algebraic expressions involves reducing them to their simplest form. This is done by performing operations such as addition, subtraction, multiplication, division, and applying rules of exponents and other algebraic principles.
In the given exercise, we simplified the algebraic expression \( \frac{x^{12} y^{13}}{x^5 y^7} \) by using the quotient rule for exponents. This allowed us to rewrite it in a simpler form: \( x^7 y^6 \). Simplifying expressions is crucial because it makes them easier to work with, whether you're solving equations, evaluating expressions, or understanding relationships between variables.
When simplifying, always look to:
Simplifying algebraic expressions involves reducing them to their simplest form. This is done by performing operations such as addition, subtraction, multiplication, division, and applying rules of exponents and other algebraic principles.
In the given exercise, we simplified the algebraic expression \( \frac{x^{12} y^{13}}{x^5 y^7} \) by using the quotient rule for exponents. This allowed us to rewrite it in a simpler form: \( x^7 y^6 \). Simplifying expressions is crucial because it makes them easier to work with, whether you're solving equations, evaluating expressions, or understanding relationships between variables.
When simplifying, always look to:
- Identify and utilize the relevant algebraic rules and properties.
- Break down the expression into smaller, manageable parts.
- Always check that each step follows logically from the last.
Other exercises in this chapter
Problem 100
Evaluate each expression using exponential rules. Write each result in standard form. $$ \left(5 \times 10^{6}\right)\left(4 \times 10^{-8}\right) $$
View solution Problem 100
In your own words, describe the different methods that can be used to find the product: \((5 x+1)^{2}\).
View solution Problem 101
Evaluate each expression using exponential rules. Write each result in standard form. $$ \frac{8 \times 10^{-1}}{16 \times 10^{5}} $$
View solution Problem 101
Suppose that a classmate asked you why \((2 x+1)^{2}\) is not \(\left(4 x^{2}+1\right)\). Write down your response to this classmate.
View solution