Problem 41
Question
For the following problems, solve for the indicated variable. $$ k^{2}=m^{2} n^{2}, \text { for } k $$
Step-by-Step Solution
Verified Answer
Question: Given the equation \(k^2 = m^2n^2\), solve for \(k\).
Answer: \(k = mn\)
1Step 1: Identify the given equation
We are given the equation:
$$
k^2 = m^2 n^2
$$
2Step 2: Isolate k
To isolate \(k\), we need to take the square root of both sides of the equation. This will give us:
$$
\sqrt{k^2} = \sqrt{m^2 n^2}
$$
3Step 3: Simplify the equation
Now, we can simplify the equation, as the square root and the square will cancel each other out:
$$
k = \sqrt{m^2 n^2}
$$
4Step 4: Apply the product property of square roots
Recall that the product property of square roots states that \(\sqrt{ab} = \sqrt{a}\sqrt{b}\). Using this property, we can rewrite the equation as:
$$
k = \sqrt{m^2} \sqrt{n^2}
$$
5Step 5: Final calculation
Now, we can simplify the equation further, as the square root and the square will cancel each other out again:
$$
k = mn
$$
So the final solution for \(k\) is:
$$
k = mn
$$
Key Concepts
Isolate VariablesSquare RootsProduct Property of Square Roots
Isolate Variables
In algebra, one of the fundamental techniques is to isolate the variable that you're solving for. This simply means rearranging an equation so your variable is on one side and everything else is on the other. It's the process of making a variable the subject of the formula.
Take for instance our given equation:
\begin{align*} k^2 &= m^2 n^2\br>
Take for instance our given equation:
\begin{align*} k^2 &= m^2 n^2\br>
Square Roots
The square root is a mathematical concept that is the inverse operation of squaring a number. When you square a number, you multiply it by itself (\(x^2 = x * x\)). Conversely, taking the square root of a number asks the question: 'What number, when multiplied by itself, gives me this value?'
For instance, the square root of 25 is 5, because 5 multiplied by 5 equals 25 (\(\sqrt{25} = 5\)). When dealing with variables, taking the square root can help simplify an expression. From our exercise:
\begin{align*}\sqrt{k^2} &= \sqrt{m^2 n^2}\br>
For instance, the square root of 25 is 5, because 5 multiplied by 5 equals 25 (\(\sqrt{25} = 5\)). When dealing with variables, taking the square root can help simplify an expression. From our exercise:
\begin{align*}\sqrt{k^2} &= \sqrt{m^2 n^2}\br>
Product Property of Square Roots
The product property of square roots is a handy tool that states you can take the square root of a product of numbers just as you would take the square root of each number individually, and then multiply the roots. It is written as \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\).
In the context of our problem:
\begin{align*} k &= \sqrt{m^2} \cdot \sqrt{n^2}\br>
In the context of our problem:
\begin{align*} k &= \sqrt{m^2} \cdot \sqrt{n^2}\br>
Other exercises in this chapter
Problem 41
Find the equation of the line that passes through the points (1,-2) and (0,4) .
View solution Problem 41
For the following problems, solve the equations, if possible. $$ a^{2}-81=0 $$
View solution Problem 41
For the following problems, use the zero-factor property to solve the equations. $$ x(x+8)=0 $$
View solution Problem 42
For the following problems, solve the equations using the quadratic formula. $$ x^{2}+6 x+8=-x-2 $$
View solution