Problem 18 - Elementary Algebra

Question

Solve. $$ x \sqrt{3}=5 $$

Step-by-Step Solution

Answer

\( x = \frac{5 \sqrt{3}}{3} \)

Step 1: Isolate the Square Root Expression

The given equation is \( x \sqrt{3} = 5 \). We need to isolate \( x \) to solve for it. This can be done by dividing both sides of the equation by \( \sqrt{3} \).

Step 2: Divide Both Sides by \( \sqrt{3} \)

Divide both sides of the equation by \( \sqrt{3} \) to solve for \( x \): \[ x = \frac{5}{\sqrt{3}} \]

Step 3: Rationalize the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{3} \):\[ x = \frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5 \sqrt{3}}{3} \]

Step 4: Simplified Solution

After rationalizing the denominator, the simplified expression for \( x \) is:\[ x = \frac{5 \sqrt{3}}{3} \]

Key Concepts

Rationalize the Denominator

When you come across a fraction with a square root in the denominator, it's often a good idea to rationalize it. This means removing the square root by multiplying the numerator and the denominator by the same square root value.
For example, if you have the expression \( \frac{5}{\sqrt{3}} \), you would multiply both the top and bottom by \( \sqrt{3} \) to eliminate the square root from the denominator.
Here's how it works:

• Multiply \( \frac{5}{\sqrt{3}} \) by \( \frac{\sqrt{3}}{\sqrt{3}} \).
• This results in \( \frac{5 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} \).
• The denominator simplifies to 3 (since \( \sqrt{3} \times \sqrt{3} = 3 \)).
• This gives you \( \frac{5\sqrt{3}}{3} \).

By rationalizing the denominator, you simplify the expression and make it easier to work with in subsequent mathematical processes.

Square Root

The square root is a mathematical function that undoes the squaring of a number. It is represented by the symbol \( \sqrt{} \).
Taking the square root of a number is like asking "which number multiplied by itself gives me this number?".
For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \).

• Square roots are handy when dealing with equations involving exponents.
• They can be tricky in fractions, often requiring you to rationalize denominators.

In equations like \( x\sqrt{3} = 5 \), the square root appears as a multiplier, and you can isolate variables by manipulating this part of the equation appropriately.

Isolate Variable

Isolating the variable is an essential step in solving equations. It means getting the unknown variable by itself on one side of the equation so you can easily see what it equals.
To isolate \( x \) in the equation \( x\sqrt{3} = 5 \), follow these steps:

• Divide both sides of the equation by \( \sqrt{3} \), which cancels it on the side with \( x \).
• This results in \( x = \frac{5}{\sqrt{3}} \).

At this point, \( x \) has been isolated, meaning it stands alone on one side, making the solution clearer. From here, further steps like rationalizing the denominator might be necessary depending on the equation's requirements.
Remember, isolating variables is crucial for determining their values effectively.