Solve for x square root of 11x=3 square root of x+2
The problem presented is a mathematical equation in which you are asked to find the value of the variable x that satisfies the equation. The equation involves square roots and is set up as follows:
square root of (11x) equals 3 times the square root of (x + 2),
which can be written symbolically as:
√(11x) = 3√(x + 2).
To solve this problem, one would typically isolate the variable x, perhaps by squaring both sides of the equation to eliminate the square root symbols, and then solving for x using appropriate algebraic manipulation. The solution process would include ensuring that any solutions found are valid by checking for extraneous solutions that may arise from the squaring process.
$\sqrt{11 x} = 3 \sqrt{x + 2}$
Answer
Solution:
Step 1
Square both sides to eliminate the square roots: $(\sqrt{11x})^2 = (3\sqrt{x+2})^2$.
Step 2
Expand and simplify both sides of the equation.
Step 2.1
Express $\sqrt{11x}$ as $(11x)^{\frac{1}{2}}$: $((11x)^{\frac{1}{2}})^2 = (3\sqrt{x+2})^2$.
Step 2.2
Simplify the left side by applying exponent rules.
Step 2.2.1
Apply the power of a power rule: $((11x)^{\frac{1}{2}})^2$.
Step 2.2.1.1
Multiply the exponents: $(11x)^{\frac{1}{2} \cdot 2}$.
Step 2.2.1.1.1
Use the rule $(a^m)^n = a^{mn}$: $(11x)^{\frac{1}{2} \cdot 2}$.
Step 2.2.1.1.2
Simplify by cancelling out the 2 in the exponent: $(11x)^{\frac{1}{\cancel{2}} \cdot \cancel{2}}$.
Step 2.2.1.1.2.1
Resulting in the simplified expression: $11x = (3\sqrt{x+2})^2$.
Step 2.3
Simplify the right side.
Step 2.3.1
Square the term $3\sqrt{x+2}$.
Step 2.3.1.1
Apply the product rule: $11x = 3^2(\sqrt{x+2})^2$.
Step 2.3.1.1.1
Square the number 3: $11x = 9(\sqrt{x+2})^2$.
Step 2.3.1.2
Express $(\sqrt{x+2})^2$ as $(x+2)$.
Step 2.3.1.2.1
Rewrite the square root: $11x = 9((x+2)^{\frac{1}{2}})^2$.
Step 2.3.1.2.2
Apply the power of a power rule: $11x = 9(x+2)^{\frac{1}{2} \cdot 2}$.
Step 2.3.1.2.3
Simplify the exponent: $11x = 9(x+2)^{\frac{2}{2}}$.
Step 2.3.1.2.4
Cancel out the common factor: $11x = 9(x+2)^{\frac{\cancel{2}}{\cancel{2}}}$.
Step 2.3.1.2.5
Resulting in the simplified expression: $11x = 9(x+2)$.
Step 2.3.1.3
Distribute the 9: $11x = 9x + 18$.
Step 3
Isolate $x$ to solve the equation.
Step 3.1
Move $x$ terms to one side: $11x - 9x = 18$.
Step 3.2
Divide to solve for $x$.
Step 3.2.1
Divide both sides by 2: $\frac{2x}{2} = \frac{18}{2}$.
Step 3.2.2
Simplify the left side: $x = \frac{18}{2}$.
Step 3.2.3
Simplify the right side to find the solution: $x = 9$.
Knowledge Notes:
The solution involves several algebraic rules and properties:
Square Root and Exponentiation: Squaring both sides of an equation is a common technique to eliminate square roots. The square of a square root, $(\sqrt{a})^2$, simplifies to $a$.
Exponent Rules: The power of a power rule states that $(a^m)^n = a^{mn}$. When simplifying expressions with exponents, this rule is often used.
Simplifying Exponents: When an exponent is 1, it is typically omitted since any number to the power of 1 is itself.
Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses. For example, $a(b + c) = ab + ac$.
Combining Like Terms: When solving equations, it's important to combine like terms to simplify the equation. Like terms are terms that have the same variables raised to the same power.
Isolating the Variable: To solve for a variable, you need to get the variable on one side of the equation and the constants on the other. This often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
Checking Solutions: After finding a solution, it's a good practice to check it by substituting it back into the original equation to ensure it satisfies the equation.
