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Chapter 3: Problem 14

Solve each equation. If it has no solution, write "no solution." $$ \sqrt{x}=36 $$

Short Answer

Expert verified
x = 1296

Step by step solution

01

Understand the square root equation

We are given the equation \( \sqrt{x} = 36 \). Our goal is to solve for \( x \) such that this equation holds true.
02

Square both sides of the equation

To eliminate the square root, both sides of the equation need to be squared. This gives: \[ \left(\sqrt{x}\right)^2 = 36^2 \ \ \ \Rightarrow \ x = 36^2 \ \Rightarrow \ x = 1296 \ \]
03

Verify the Solution

Substitute \(x = 1296\) back into the original equation to verify the solution: \[ \sqrt{1296} = 36 \ \Rightarrow \ 36 = 36 \ \ \ \Rightarrow \ \text{The solution is correct.} \ \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebra
Algebra is the branch of mathematics that deals with symbols and rules for manipulating those symbols. In solving equations, especially square root equations, understanding algebraic principles is key. Symbols like \(x\) are used to represent unknown values. Through manipulations鈥攍ike squaring both sides of an equation鈥攚e can isolate these unknowns and solve for them.

When you see an equation such as \( \sqrt{x} = 36 \), the symbol \(x\) represents the value we want to find. Our goal is to isolate \(x\). We do this through algebraic manipulations which we will delve deeper into in the following sections.
Square Root
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 36 (\( \sqrt{36}\)) is 6, because \(6 \times 6 = 36\).

In our exercise, \( \sqrt{x} = 36\), we are given a square root equation and need to find the number \(x\) such that its square root is 36. To solve this, we use the property of square roots where squaring both sides of the equation eliminates the square root.

When we square both sides:
\[ \left( \sqrt{x} \right)^{2} = 36^{2} \]
The square root and the square cancel each other out on the left side, leaving us with:
\[ x = 36^{2} \]
\[ x = 1296 \]
Hence, \(x\) must be 1296.
Equation Solving
Solving equations means finding all values of the unknowns that make the equation true. For the given square root equation \( \sqrt{x} = 36\), the steps are as follows:
  • First, identify what you need to isolate鈥攊n this case, \(x\).

  • To eliminate the square root, square both sides of the equation:

    • \[ \left( \sqrt{x} \right)^{2} = 36^{2} \]
  • The square root and the square cancel each other:

    • \[ x = 36^{2} \]
  • Calculate the result:

    • \[ x = 1296 \]

Verification is the final step to ensure that our solution is correct. Substitute \(x = 1296\) back into the original equation to check:
\[ \sqrt{1296} = 36 \]
\[ 36 = 36 \]
The left-hand side equals the right-hand side, confirming that our solution \(x = 1296\) is accurate. This process underscores the importance of each algebraic transformation and verification step in solving equations.

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Most popular questions from this chapter

Write a quadratic equation for each table. $$ \begin{array}{r|r}{x} & {y} \\ \hline-3 & {18} \\ {-2} & {8} \\ {-1} & {2} \\\ {0} & {0} \\ {1} & {2} \\ {2} & {8} \\ {3} & {18} \\ \hline\end{array} $$

Rewrite each expression using a single base and a single exponent. $$\left(22^{2} \cdot 22^{5}\right)^{0}$$

Solve each equation. If it has no solution, write "no solution." $$ \sqrt{x-3}=9 $$

Tell whether each number is rational or irrational. If is rational, find two integers whose ratio is equal to it. If it is irrational, explain how you know. $$ 4 \frac{1}{3} $$

In Exercises 28鈥�35, find the indicated roots without using a calculator. the cube root of \(-216\)
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