Solve each equation. See Examples 4–6. √(3√(2x+3)) = √(5x-6)

Verified step by step guidance

1. Step 1: Start by squaring both sides of the equation to eliminate the outer square roots: $(\sqrt{3\sqrt{2x+3}}{)}^2=(\sqrt{5x-6}{)}^2$.
2. Step 2: Simplify both sides after squaring: $3\sqrt{2x+3}=5x-6$.
3. Step 3: Isolate the inner square root by dividing both sides by 3: $\sqrt{2x+3}=\frac{5x-6}{3}$.
4. Step 4: Square both sides again to eliminate the remaining square root: $(\sqrt{2x+3}{)}^2={\left(\frac{5x-6}{3}\right)}^2$.
5. Step 5: Simplify the equation: $2x+3=\frac{(5x-6{)}^2}{9}$.

Key Concepts

Square Roots

Square roots are mathematical operations that find a number which, when multiplied by itself, gives the original number. In the equation √(3√(2x+3)) = √(5x-6), understanding how to manipulate square roots is essential for isolating variables and simplifying expressions. This concept is foundational in solving equations involving radicals.

Radical Equations

Radical equations are equations that contain a variable within a radical (square root, cube root, etc.). To solve these equations, one typically squares both sides to eliminate the radical, but this can introduce extraneous solutions. Recognizing how to handle these equations is crucial for finding valid solutions.

Isolating Variables

Isolating variables involves rearranging an equation to get the variable of interest on one side. This process often requires using inverse operations, such as addition, subtraction, multiplication, or division. In the context of the given equation, isolating x is necessary to find its value after simplifying the radicals.