Evaluate square root of x^2 = square root of 36
The question provided asks you to determine the value of 'x' by comparing the square root of 'x' squared to the square root of 36. Essentially, you are being asked to solve an equation where you will need to apply properties of square roots and possibly deal with the absolute value concept depending on the interpretation of the square root of 'x' squared. It's a problem that requires knowledge of algebra and square root properties, and it involves understanding that the principal square root of a nonnegative number is always nonnegative.
$\sqrt{x^{2}} = \sqrt{36}$
Answer
Solution:
Step 1:
Square both sides to eliminate the square roots: $(\sqrt{x^2})^2 = (\sqrt{36})^2$.
Step 2:
Proceed to simplify the equation.
Step 2.1:
Rewrite the square root of $x^2$ using the property $\sqrt[n]{a^x} = a^{\frac{x}{n}}$: $(x^{\frac{2}{2}})^2 = (\sqrt{36})^2$.
Step 2.2:
Simplify the exponent by dividing $2$ by $2$: $(x^1)^2 = (\sqrt{36})^2$.
Step 2.3:
Focus on simplifying the left-hand side.
Step 2.3.1:
Apply the exponent multiplication rule: $(x^1)^2$.
Step 2.3.1.1:
Use the power rule $(a^m)^n = a^{mn}$: $x^{1 \cdot 2} = (\sqrt{36})^2$.
Step 2.3.1.2:
Calculate the product of $2$ and $1$: $x^2 = (\sqrt{36})^2$.
Step 2.4:
Now, simplify the right-hand side.
Step 2.4.1:
Simplify the squared square root: $(\sqrt{36})^2$.
Step 2.4.1.1:
Express $36$ as $6^2$: $x^2 = (\sqrt{6^2})^2$.
Step 2.4.1.2:
Extract terms from under the radical, assuming they represent positive real numbers: $x^2 = 6^2$.
Step 2.4.1.3:
Raise $6$ to the power of $2$: $x^2 = 36$.
Step 3:
Solve for $x$.
Step 3.1:
Take the square root of both sides to remove the exponent: $x = \pm \sqrt{36}$.
Step 3.2:
Simplify the square root of $36$.
Step 3.2.1:
Represent $36$ as $6^2$: $x = \pm \sqrt{6^2}$.
Step 3.2.2:
Extract terms from under the radical: $x = \pm 6$.
Step 3.3:
Combine the positive and negative solutions to complete the solution.
Step 3.3.1:
Use the positive value from $\pm$ for the first solution: $x = 6$.
Step 3.3.2:
Use the negative value from $\pm$ for the second solution: $x = -6$.
Step 3.3.3:
State the full solution with both positive and negative values: $x = 6, -6$.
Knowledge Notes:
Square Roots and Exponents: The square root of a number $a$, denoted as $\sqrt{a}$, is a value that, when multiplied by itself, gives $a$. The square of a number is represented by an exponent of $2$. The square root and squaring are inverse operations.
Properties of Exponents: The power rule states that $(a^m)^n = a^{mn}$. This is used when raising a power to another power.
Simplifying Square Roots: When a square root is squared, such as $(\sqrt{a})^2$, the result is simply $a$. This is because the square root and the square cancel each other out.
Solving Quadratic Equations: When solving equations involving squares, such as $x^2 = a$, the solutions can be both positive and negative, represented as $x = \pm \sqrt{a}$.
Assumption of Positive Real Numbers: When extracting terms from under a square root during simplification, it is often assumed that the terms represent positive real numbers unless otherwise specified. This is because the square root function is conventionally taken to return the principal (non-negative) root.
