Solve for h 4 = square root of h-6

Solution:

Step 1:

Express the equation in its standard form: $4 = \sqrt{h - 6}$.

Step 2:

Eliminate the square root by squaring both sides: $(\sqrt{h - 6})^2 = 4^2$.

Step 3:

Expand and simplify the equation.

Step 3.1:

Represent the square root as a power: $(h - 6)^{\frac{1}{2}}$.

Step 3.2:

Simplify the left-hand side.

Step 3.2.1:

Square the expression with the fractional exponent: $((h - 6)^{\frac{1}{2}})^2$.

Step 3.2.1.1:

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

Step 3.2.1.1.1:

Multiply the exponents: $(h - 6)^{\frac{1}{2} \cdot 2} = 4^2$.

Step 3.2.1.1.2:

Reduce the exponents: $(h - 6)^{1} = 4^2$.

Step 3.2.1.2:

Simplify the expression: $h - 6 = 4^2$.

Step 3.3:

Simplify the right-hand side: $h - 6 = 16$.

Step 4:

Isolate the variable $h$.

Step 4.1:

Add 6 to both sides: $h = 16 + 6$.

Step 4.2:

Combine like terms: $h = 22$.

Knowledge Notes:

The problem-solving process involves solving an equation with a square root. The key knowledge points include:

1. Square Roots: The square root of a number $x$ is a value that, when multiplied by itself, gives $x$. The square root function is the inverse of squaring a number.

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

3. Exponent Rules: When raising a power to a power, you multiply the exponents. For example, $(a^m)^n = a^{mn}$.

4. Simplifying Equations: After removing the square root, the equation can be simplified by combining like terms and isolating the variable.

5. Isolating the Variable: To solve for a variable, we need to get the variable on one side of the equation and all other terms on the opposite side. This often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number.

By following these steps and applying these principles, one can solve equations involving square roots and isolate the variable of interest.