Last week Sherri bought 324 square yards of sod to grass a square play area for the children of her day care. Now she has to fence the area to keep the children safe but does not know how many yards of fencing she needs to buy. All Sherri can figure out is that all the sides are the same length because the grassy area is a square.

How can she determine how many yards of fencing to buy?

In this concept, you will learn to solve equations involving radicals.

### Radicals

When you solve an equation you are trying to find the value for the variable that will make the equality statement true. The steps applied to solving an equation are inverse operations. To solve an equation involving radicals, inverse operations are used to solve for the variable.

A radical involving the square root of a number can be evaluated by determining the square root of the number under the radical sign. If the radicand is a perfect square then its square will be the number which multiplied by itself twice will give the value of the radicand. For example the square root of 81 can be denoted by . What number times itself twice gives 81?

Taking the square root of a number is the inverse operation of squaring and vice versa.

Let’s look at an equation involving radicals.

The variable ‘squared and its value is 121. To solve this equation the value of needs to be determined. The inverse operation of squaring is taking the square root. Remember, whatever operation is applied to one side of the equation must also be applied to the other side.

’ isFirst, take the square root of both sides of the equation.

Next, verify the answer by substituting the value for ‘

’ into the original equation.

Then, perform any indicated operations.

The answer is 11.

The value of 11 made the equality statement true – both sides of the equation are the same.

Let’s look at one more.

Solve the following equation involving radicals:

\begin{align*}\sqrt{x+2}=6\end{align*}

Notice the left side of the equation is a radical. The inverse operation of taking the square root is squaring.

Remember, the square of the square root of anything is the anything. In other words

First, square both sides of equation.

Next, perform any indicated operations and simplify the equation.

Then, subtract 2 from both sides of the equation to solve the equation for ‘

’.

Next, verify the answer by substituting the value for ‘\begin{align*}x\end{align*}’ into the original equation.

Then, perform any indicated operations and simplify the equation.

The answer is 34.

The value of 34 made the equation true.

### Examples

#### Example 1

Earlier, you were given a problem about Sherri and the fence for the grassy square. She needs to figure out the perimeter of the square.

First, write an equation for the area of the grassy square.

\begin{align*}A = s^2\end{align*}

Such that \begin{align*}A\end{align*} is the area and \begin{align*}s\end{align*} is the length of any side of the square.

Next, fill in the value for the area.

Next, solve the equation for ‘

’ by taking the square root of both sides of the radical equation.

The answer is 18.

The length of each side of the square is 18 yds.

Next, write an equation for finding the perimeter of the square. The perimeter is the distance all around the grassy area.

Such that ‘

’ is the perimeter and ‘ ’ is the length of one side of the square.Next, fill in the value for ‘

’ and perform the indicated operation.

The answer is 72.

Sherri needs to buy 72 yards of fencing.

#### Example 2

Solve the following equation involving radicals to the nearest tenth.

\begin{align*}x^2 = 26\end{align*}

First, the opposite operation of squaring is taking the square root. Take the square root of both sides of the equation.

Notice that the number 26 is not a perfect square. Use the TI calculator to find the square root of 26.

Next, on the calculator press 2^{nd} enter.

Then, round the value shown on the screen to one place after the decimal.

The answer is 5.1

#### Example 3

Solve the following radical equation for the variable ‘

’.

First, take the square root of both sides of the equation.

Next, simplify both sides of the equation.

The answer is 12.

#### Example 4

Solve the following radical equation for the variable ‘

’.\begin{align*}\sqrt{a-8} = 7\end{align*}

First, apply the inverse operation of taking the square root to both sides of the equation.

Next, perform any indicated operations and simplify the equation.

Then, add eight to both sides of the equation to solve for the variable ‘

’.

Next, verify the answer by substituting the value for ‘

’ into the original equation.

Then, perform any indicated operations and simplify the equation.

The answer is 57.

#### Example 5

Solve the following equation for the variable.

First, isolate the variable by subtracting 4 from both sides of the equation.

\begin{align*}\begin{array}{rcl} x^3+4 &=& 31\\ x^3+4-4 &=& 31-4 \end{array}\end{align*}

Next, simplify both sides of the equation.

Next, perform the inverse operation of cubing – taking the cube root, on both sides of the equation.

Then, perform the indicated operations.

The answer is 3.

### Review

Solve each equation involving radical expressions.

- \begin{align*}x^2 = 169\end{align*}
- \begin{align*}x^2 = 16\end{align*}
- \begin{align*}x^3 = 64\end{align*}
- \begin{align*}x^3 = 27\end{align*}
- \begin{align*}x^2+3 = 147\end{align*}
- \begin{align*}x^3+10 = 135\end{align*}
- \begin{align*}x^2-4 = 21\end{align*}
- \begin{align*}x^2-6 = 30\end{align*}

### Review (Answers)