Problem 5
Question
Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=\left(3 x^{5}\right)^{2}$$
Step-by-Step Solution
Verified Answer
Yes, it is a power function: \( y = 9x^{10} \), with \( k = 9 \) and \( p = 10 \).
1Step 1: Simplify the Expression
First, simplify the given expression \( y = (3x^5)^2 \). To do this, apply the power of a product and power of a power rules: \( a^m \cdot a^n = a^{m+n}\) and \((a^m)^n = a^{m \cdot n} \). Thus, \((3x^5)^2 = 3^2 \cdot (x^5)^2 = 9 \cdot x^{10} \).
2Step 2: Identify as a Power Function
Next, determine if the simplified expression \( y = 9x^{10} \) is a power function. A function is a power function if it can be written in the form \( y = kx^p \), where \( k \) and \( p \) are constants. In this expression, \( y = 9x^{10} \) fits the form and is thus a power function.
3Step 3: Identify Constants \(k\) and \(p\)
In the expression \( y = 9x^{10} \), compare it to the general power function form \( y = kx^p \). Here, the constant \( k = 9 \) and the exponent \( p = 10 \).
4Step 4: Confirm the Form
Confirm that the expression \( y = 9x^{10} \) can be rewritten in the form \( y = kx^p \) with \( k = 9 \) and \( p = 10 \), which matches our definition of a power function.
Key Concepts
Simplifying ExpressionsPower RulesIdentifying Constants
Simplifying Expressions
Whenever you encounter a mathematical expression, your first step should be simplification. Simplifying expressions makes them easier to work with and understand. In the example of \( y = (3x^5)^2 \), we aim to break it down by applying various mathematical rules.
One important rule is the **power of a power rule**. This states that \((a^m)^n = a^{m \cdot n}\). For our expression, that means taking each part separately. The base \(3\) is raised to the power 2, and so is the base \(x^5\). Thus, \(3^2 = 9\) and \((x^5)^2 = x^{10}\).
Combining these results, we arrive at the simplified expression \( y = 9x^{10} \). This simplification makes it more manageable and leads directly to identifying types and constants.
One important rule is the **power of a power rule**. This states that \((a^m)^n = a^{m \cdot n}\). For our expression, that means taking each part separately. The base \(3\) is raised to the power 2, and so is the base \(x^5\). Thus, \(3^2 = 9\) and \((x^5)^2 = x^{10}\).
Combining these results, we arrive at the simplified expression \( y = 9x^{10} \). This simplification makes it more manageable and leads directly to identifying types and constants.
Power Rules
Understanding power rules is essential for identifying power functions in expressions. A power function is a specific type of mathematical function represented by the form \( y = kx^p \), where \( k \) and \( p \) are constants. Power rules help simplify expressions and showcase whether an expression fits this form.
When examining the expression \( y = 9x^{10} \), we see it conforms perfectly to the power function form, with \( k = 9 \) and \( p = 10 \).
These rules, like the **product of powers** rule (\( a^m \cdot a^n = a^{m+n} \)) and the **power of a power** rule, allow us to manipulate and simplify expressions efficiently. So, when working with any function, these power rules are invaluable tools that simplify the process and clarify the structure.
When examining the expression \( y = 9x^{10} \), we see it conforms perfectly to the power function form, with \( k = 9 \) and \( p = 10 \).
These rules, like the **product of powers** rule (\( a^m \cdot a^n = a^{m+n} \)) and the **power of a power** rule, allow us to manipulate and simplify expressions efficiently. So, when working with any function, these power rules are invaluable tools that simplify the process and clarify the structure.
Identifying Constants
In the context of power functions, constants play a crucial role. A constant is a fixed value that doesn't change, unlike a variable which can change. When you have an expression in the form \( y = kx^p \), you'll need to identify both \( k \) and \( p \) as constants because they define your function.
For the expression \( y = 9x^{10} \), identifying constants means figuring out these fixed numbers. Here, we can see that the "9" in front of \( x^{10} \) is our constant \( k = 9\).
Additionally, the exponent "10" in \( x^{10} \) is another constant, which we call \( p = 10\). By identifying these constants, you confirm that the expression is indeed a power function. Recognizing the constants not only verifies its structure but also makes it easier to predict behavior, graph functions, and understand their broader applications.
For the expression \( y = 9x^{10} \), identifying constants means figuring out these fixed numbers. Here, we can see that the "9" in front of \( x^{10} \) is our constant \( k = 9\).
Additionally, the exponent "10" in \( x^{10} \) is another constant, which we call \( p = 10\). By identifying these constants, you confirm that the expression is indeed a power function. Recognizing the constants not only verifies its structure but also makes it easier to predict behavior, graph functions, and understand their broader applications.
Other exercises in this chapter
Problem 4
For the functions \(f\) and \(g\), find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=\sqrt{x+4}, g(x)=x^{2}$$
View solution Problem 5
Sketch graphs of the functions. What are their amplitudes and periods? $$y=3 \sin 2 x$$
View solution Problem 5
If a bank pays \(6 \%\) per year interest compounded continuously, how long does it take for the balance in an account to double?
View solution Problem 5
Solve for \(t\) using natural logarithms. $$50=10 \cdot 3^{t}$$
View solution