Problem 7
Question
Use the quotient of powers property to simplify the expression. $$ \frac{x^{12}}{x^{9}} $$
Step-by-Step Solution
Verified Answer
After applying the quotient of powers property and simplifying, \( \frac{x^{12}}{x^{9}} = x^3 \).
1Step 1: Identify the bases and their exponents
Here the base is \(x\) in both terms of the fraction. The exponent in the numerator is 12 and in the denominator is 9.
2Step 2: Applying the Quotient of Powers Property
According to the quotient of powers property, when we divide like bases we subtract the exponents. So, \(x^{12} / x^{9} = x^{12-9}\).
3Step 3: Simplify the exponent
Subtract the exponents. 12 - 9 equals 3. So the simplified version is \(x^3\).
Key Concepts
Simplifying ExpressionsExponent RulesAlgebraic Fractions
Simplifying Expressions
Simplifying expressions is like tidying up a messy room so it's easier to see exactly what is there. In algebra, it involves combining like terms, reducing fractions, and applying various algebraic laws to make expressions as straightforward as possible. Think of it as a way to make an expression more user-friendly: It might not change the value, but it certainly makes it easier to work with!
In the case of the exercise involving \(\frac{x^{12}}{x^{9}}\), simplifying the expression means we need to find the simplest form that represents the same quantity. It is important to remember that we do this to prevent mistakes and to make further calculations simpler. For example, \(x^3\) is much easier to comprehend and use in subsequent equations than dealing with the original fraction.
In the case of the exercise involving \(\frac{x^{12}}{x^{9}}\), simplifying the expression means we need to find the simplest form that represents the same quantity. It is important to remember that we do this to prevent mistakes and to make further calculations simpler. For example, \(x^3\) is much easier to comprehend and use in subsequent equations than dealing with the original fraction.
Exponent Rules
When it comes to calculating with powers, certain rules or 'shortcuts'—known as exponent rules—make our lives easier. These rules are the foundation for working neatly with exponents, and they include the product of powers, quotient of powers, power of a power, and more.
For the quotient of powers property, as in \(\frac{x^{12}}{x^{9}}\), you subtract the exponents when dividing like bases. This property is built on the idea that when you're dividing, you're essentially asking how many times the base in the denominator can be multiplied to reach the base in the numerator. Subtracting the exponents gives you the number of times the base must be multiplied by itself to bridge this gap. Hence, \(x^{12} / x^{9} = x^{12-9} = x^3\).
It's crucial to grasp these rules well because they serve as keystones for more advanced mathematics including calculus and beyond.
For the quotient of powers property, as in \(\frac{x^{12}}{x^{9}}\), you subtract the exponents when dividing like bases. This property is built on the idea that when you're dividing, you're essentially asking how many times the base in the denominator can be multiplied to reach the base in the numerator. Subtracting the exponents gives you the number of times the base must be multiplied by itself to bridge this gap. Hence, \(x^{12} / x^{9} = x^{12-9} = x^3\).
It's crucial to grasp these rules well because they serve as keystones for more advanced mathematics including calculus and beyond.
Algebraic Fractions
Algebraic fractions are just like ordinary fractions, but instead of numbers, they have variables like \( x \) or \( y \) in their numerators and denominators. Understanding them is essential for solving algebraic equations, especially when you come across expressions that involve division by variables.
In the context of the given exercise, \(\frac{x^{12}}{x^{9}}\) is an algebraic fraction. To handle such fractions, you apply the same principles you would to numerical fractions—like simplifying them to their lowest terms. Here, given that they share the same base, we simplify using the quotient of powers property. What's left is an algebraic expression (\(x^3\)) without the fraction form, indicating that such fractions can often be simplified to whole expressions that are easier to use in further calculations.
In the context of the given exercise, \(\frac{x^{12}}{x^{9}}\) is an algebraic fraction. To handle such fractions, you apply the same principles you would to numerical fractions—like simplifying them to their lowest terms. Here, given that they share the same base, we simplify using the quotient of powers property. What's left is an algebraic expression (\(x^3\)) without the fraction form, indicating that such fractions can often be simplified to whole expressions that are easier to use in further calculations.
Other exercises in this chapter
Problem 7
Identify the initial amount and the growth rate in the exponential function. $$y=100(1+0.5)^{t}$$
View solution Problem 7
Write the number in decimal form. $$ 2.45 \times 10^{-1} $$
View solution Problem 7
Tell whether the graph of the function contains the point \((0,1) .\) Explain your answer. $$y=2^{x}$$
View solution Problem 7
Use the power of a power property to write the expression as a single power of the base. \(\left(2^{4}\right)^{3}\)
View solution