Solve for x square root of x-5-7=0
The problem is asking to find the value of the variable x that satisfies the equation involving a square root: \(\sqrt{x - 5} - 7 = 0\). This equation implies that there is an expression under the square root which, when simplified, will allow us to isolate x and determine its value. The problem involves algebraic manipulation, including undoing the square root by squaring both sides of the equation at an appropriate stage, in order to solve for x.
$\sqrt{x - 5 - 7} = 0$
Answer
Solution:
Step:1
To isolate the square root, we will square both sides of the equation. $(\sqrt{x - 5} - 7)^{2} = 0^{2}$
Step:2
Proceed to simplify the equation.
Step:2.1
Express the square root as a power: $\sqrt{x - 5} = (x - 5)^{\frac{1}{2}}$. $(x - 5)^{\frac{1}{2} - 7})^{2} = 0^{2}$
Step:2.2
Focus on simplifying the left-hand side.
Step:2.2.1
Simplify the squared expression.
Step:2.2.1.1
Apply the exponent rules to multiply the exponents. $(x - 5)^{\frac{1}{2} \cdot 2} - 7^{2} = 0^{2}$
Step:2.2.1.1.1
Use the power rule $(a^{m})^{n} = a^{mn}$. $(x - 5)^{1} - 7^{2} = 0^{2}$
Step:2.2.1.1.2
Eliminate the common exponent of $2$.
Step:2.2.1.1.2.1
Remove the exponent by simplification. $(x - 5) - 7^{2} = 0^{2}$
Step:2.2.1.1.2.2
Rewrite the equation. $x - 5 - 49 = 0^{2}$
Step:2.2.1.2
Combine like terms. $x - 54 = 0^{2}$
Step:2.2.1.3
Simplify the equation. $x - 54 = 0$
Step:2.3
Simplify the right-hand side.
Step:2.3.1
Recognize that any non-negative number raised to the power of $0$ is $1$. $x - 54 = 0$
Step:3
Add $54$ to both sides to solve for $x$. $x = 54$
Knowledge Notes:
Square Roots and Exponents: The square root of a number can be represented as that number raised to the power of $\frac{1}{2}$. This is based on the general rule that $\sqrt[n]{a} = a^{\frac{1}{n}}$.
Squaring Both Sides: When dealing with equations involving square roots, one common technique is to square both sides of the equation to eliminate the square root.
Power Rule: The power rule for exponents states that $(a^{m})^{n} = a^{m \cdot n}$. This is used to simplify expressions where an exponent is raised to another exponent.
Simplifying Equations: The process of simplifying equations involves combining like terms, applying arithmetic operations, and reducing expressions to their simplest form.
Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to one, i.e., $a^{0} = 1$ for $a \neq 0$. This rule is used to simplify expressions where zero is an exponent.
Solving Linear Equations: To solve linear equations, we perform operations that will isolate the variable on one side of the equation. This often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
