Evaluate square root of x+2=5
Explanation: The given problem is asking to solve an equation where the square root of a quantity (x+2) is equal to 5. This problem is in the realm of algebra and requires you to isolate and find the value of the variable x that will make the equation true. You would typically proceed by squaring both sides of the equation to eliminate the square root and then simplify the resulting expression to solve for x.
$\sqrt{x + 2} = 5$
Answer
Solution:
Step 1:
Square both sides to eliminate the square root: $(\sqrt{x + 2})^2 = 5^2$.
Step 2:
Expand and simplify the equation.
Step 2.1:
Express the square root as a power: $(x + 2)^{\frac{1}{2}}$.
Step 2.2:
Simplify the left-hand side.
Step 2.2.1:
Apply the exponent rules to $(x + 2)^{\frac{1}{2} \cdot 2}$.
Step 2.2.1.1:
Use the rule $(a^m)^n = a^{m \cdot n}$.
Step 2.2.1.1.1:
Multiply the exponents: $(x + 2)^{\frac{1}{2} \cdot 2} = 5^2$.
Step 2.2.1.1.2:
Reduce the exponents: $(x + 2)^{\frac{1}{\cancel{2}} \cdot \cancel{2}} = 5^2$.
Step 2.2.1.1.2.1:
Simplify the expression: $(x + 2)^1 = 5^2$.
Step 2.2.1.1.2.2:
Rewrite without the exponent: $x + 2 = 5^2$.
Step 2.2.1.2:
Conclude the simplification: $x + 2 = 5^2$.
Step 2.3:
Simplify the right-hand side.
Step 2.3.1:
Calculate $5^2$: $x + 2 = 25$.
Step 3:
Isolate $x$ by moving constants to the other side.
Step 3.1:
Subtract $2$ from both sides: $x = 25 - 2$.
Step 3.2:
Perform the subtraction: $x = 23$.
Knowledge Notes:
Square Roots and Exponents: The square root of a number can be represented as that number raised to the power of $\frac{1}{2}$. Squaring the square root of a number will return the original number, as the exponents cancel each other out.
Exponent Rules: When an exponent is raised to another exponent, the exponents are multiplied together. This is known as the power of a power rule, expressed as $(a^m)^n = a^{m \cdot n}$.
Solving Radical Equations: To solve equations involving square roots, one typically squares both sides to remove the radical. Afterward, the equation can be solved like any other algebraic equation.
Isolating the Variable: To solve for a variable, one must isolate it on one side of the equation. This often involves performing inverse operations to both sides of the equation, such as adding, subtracting, multiplying, or dividing, to cancel out other terms.
Checking Solutions: It's important to check the solutions in the original equation, especially when dealing with square roots, as squaring both sides can introduce extraneous solutions.
