Solve the Rational Equation for v square root of v+3-1=7
The given problem is a mathematical equation that involves solving for the variable v. The equation provided contains a rational term and a radical term. Specifically, the equation features the square root of the variable v added to 3, and this expression has been subtracted by 1 and set equal to the number 7. The challenge is to manipulate the equation algebraically to isolate the variable v and find its value(s) that satisfy the equation. This would typically involve operations such as squaring both sides to eliminate the square root, simplifying the expression, and solving a resulting polynomial equation.
$\sqrt{v + 3} - 1 = 7$
Answer
Solution:
Step 1:
Isolate the radical expression involving $v$ by shifting all other terms to the opposite side of the equation.
Step 1.1:
Increase both sides of the equation by $1$ to achieve $\sqrt{v + 3} = 7 + 1$.
Step 1.2:
Combine the constants on the right-hand side to simplify $\sqrt{v + 3} = 8$.
Step 2:
Eliminate the square root by raising both sides of the equation to the power of $2$, resulting in $\left(\sqrt{v + 3}\right)^{2} = 8^{2}$.
Step 3:
Expand and simplify both sides of the equation.
Step 3.1:
Express the square root as a power by using $\sqrt{v + 3}$ as $\left(v + 3\right)^{\frac{1}{2}}$.
Step 3.2:
Simplify the left-hand side of the equation.
Step 3.2.1:
Focus on simplifying $\left(\left(v + 3\right)^{\frac{1}{2}}\right)^{2}$.
Step 3.2.1.1:
Apply the exponent multiplication rule to $\left(\left(v + 3\right)^{\frac{1}{2}}\right)^{2}$.
Step 3.2.1.1.1:
Utilize the power of a power property, $\left(a^{m}\right)^{n} = a^{m \cdot n}$.
Step 3.2.1.1.2:
Reduce the expression by canceling out the common exponent factors.
Step 3.2.1.1.2.1:
Eliminate the common factors to simplify the expression.
Step 3.2.1.1.2.2:
Rewrite the equation without the exponents to get $v + 3 = 8^{2}$.
Step 3.2.1.2:
Reduce the equation to its simplest form, $v + 3 = 8^{2}$.
Step 3.3:
Simplify the right-hand side by calculating $8$ to the power of $2$, resulting in $v + 3 = 64$.
Step 4:
Shift all terms not including $v$ to the other side of the equation.
Step 4.1:
Decrease both sides by $3$ to isolate $v$, giving $v = 64 - 3$.
Step 4.2:
Subtract $3$ from $64$ to find the value of $v$, which is $v = 61$.
Knowledge Notes:
To solve a rational equation that involves a square root, one typically follows these steps:
Isolate the radical expression on one side of the equation.
Square both sides of the equation to eliminate the square root.
Simplify the resulting equation.
Solve for the variable.
Key knowledge points include:
Isolating the Radical: This involves moving all terms without the radical to the other side of the equation.
Squaring Both Sides: When you square both sides of an equation, the square root is eliminated because $(\sqrt{x})^2 = x$.
Simplifying the Equation: This may involve combining like terms or using exponent rules such as $(a^m)^n = a^{mn}$.
Checking Solutions: It's important to check that the solutions obtained do not result in any negative numbers under the square root when plugged back into the original equation, as this would be invalid in the real number system.
In the context of this problem, the equation is squared to remove the square root, and then basic algebraic operations are used to isolate the variable and solve for it.
