Solve for a square root of 5a+1+2=6

Solution:

Step 1:

Isolate the square root term $\sqrt{5a + 1}$ by moving other terms to the opposite side of the equation.

Step 1.1:

Subtract $2$ from both sides to get $\sqrt{5a + 1} = 6 - 2$.

Step 1.2:

Perform the subtraction to find $\sqrt{5a + 1} = 4$.

Step 2:

Square both sides to eliminate the square root, resulting in $(\sqrt{5a + 1})^2 = 4^2$.

Step 3:

Simplify both sides of the squared equation.

Step 3.1:

Express the square root as a power: $(5a + 1)^{\frac{1}{2}}$.

Step 3.2:

Simplify the left side by squaring the term.

Step 3.2.1:

Apply the exponent rule: $(a^m)^n = a^{mn}$.

Step 3.2.1.1:

Multiply the exponents to get $(5a + 1)^{\frac{1}{2} \cdot 2} = 4^2$.

Step 3.2.1.1.2:

Simplify the exponent to $(5a + 1)^1 = 4^2$.

Step 3.2.1.2:

Resolve the exponent to reveal $5a + 1 = 4^2$.

Step 3.3:

Simplify the right side by calculating $4^2$.

Step 3.3.1:

Compute the square of $4$ to get $5a + 1 = 16$.

Step 4:

Solve for the variable $a$.

Step 4.1:

Move constants to the other side to isolate the term with $a$.

Step 4.1.1:

Subtract $1$ from both sides, yielding $5a = 16 - 1$.

Step 4.1.2:

Complete the subtraction to find $5a = 15$.

Step 4.2:

Divide by the coefficient of $a$ to solve for $a$.

Step 4.2.1:

Divide $5a = 15$ by $5$ to get $\frac{5a}{5} = \frac{15}{5}$.

Step 4.2.2:

Simplify the left side by canceling out the $5$.

Step 4.2.2.1:

Reduce the fraction to find $a = \frac{15}{5}$.

Step 4.2.3:

Simplify the right side by dividing.

Step 4.2.3.1:

Calculate the division to conclude that $a = 3$.

Knowledge Notes:

To solve an equation involving a square root, the following knowledge points are relevant:

1. Isolating the Square Root: The first step in solving square root equations is to isolate the square root on one side of the equation.

2. Squaring Both Sides: To eliminate the square root, square both sides of the equation. This is because $(\sqrt{x})^2 = x$.

3. Simplifying Exponents: When simplifying expressions with exponents, the power rule $(a^m)^n = a^{mn}$ is often used. In this case, squaring a square root effectively removes the root.

4. Solving Linear Equations: After removing the square root and simplifying the equation, you are often left with a linear equation that can be solved by isolating the variable. This typically involves moving constants to the other side and dividing by the coefficient of the variable.

5. Checking Solutions: It is important to check the solutions in the original equation, as squaring both sides can introduce extraneous solutions that do not satisfy the original equation. However, in this particular problem, we only have one solution, and it does not need to be checked as the steps involve straightforward arithmetic operations.