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## Find the roots of basic equations containing roots

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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.

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 \begin{align*}\sqrt{81}\end{align*}. What number times itself twice gives 81?

\begin{align*}\begin{array}{rcl} 81 = 9 \times 9 &=& 9^2\\ \therefore \sqrt{81} &=& 9 \end{array}\end{align*}

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

Let’s look at an equation involving radicals.

\begin{align*}x^2 = 121\end{align*}

The variable ‘\begin{align*}x\end{align*}’ is 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.

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

\begin{align*}\begin{array}{rcl} x^2 &=& 121\\ \sqrt{x^2} &=& \sqrt{121}\\ x &=& 11 \end{array}\end{align*}

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

\begin{align*}\begin{array}{rcl} x^2 &=& 121\\ (11)^2 &=& 121 \end{array}\end{align*}

Then, perform any indicated operations.

\begin{align*}\begin{array}{rcl} 11 \times 11 &=& 121\\ 121 &=& 121 \end{array}\end{align*}

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

\begin{align*}\left(\sqrt{\text{anything}} \right)^2 = \text{anything}.\end{align*}

First, square both sides of equation.

\begin{align*}\begin{array}{rcl} \sqrt{x+2} &=& 6\\ \left(\sqrt{x+2} \right)^2 &=& (6)^2 \end{array}\end{align*}

Next, perform any indicated operations and simplify the equation.

\begin{align*}\begin{array}{rcl} \left(\sqrt{x+2} \right)^2 &=& x+2\\ (6)^2 = 6 \times 6 &=& 36\\ x+2 &=& 36 \end{array}\end{align*}

Then, subtract 2 from both sides of the equation to solve the equation for ‘\begin{align*}x\end{align*}’.

\begin{align*}\begin{array}{rcl} x+2 &=& 36\\ x+2-2 &=& 36-2\\ x &=& 34 \end{array}\end{align*}

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

\begin{align*}\begin{array}{rcl} \sqrt{x+2} &=& 6\\ \sqrt{{\color{blue}34}+2} &=& 6 \end{array}\end{align*}

Then, perform any indicated operations and simplify the equation.

\begin{align*}\begin{array}{rcl} \sqrt{{\color{blue}34}+2} &=& 6\\ \sqrt{36} &=& 6\\ 36 = 6 \times 6 &=& 6^2\\ 6 &=& 6 \end{array}\end{align*}

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.

\begin{align*}\begin{array}{rcl} A &=& s^2\\ 324 &=& s^2 \end{array}\end{align*}

Next, solve the equation for ‘\begin{align*}s\end{align*}’ by taking the square root of both sides of the radical equation.

\begin{align*}\begin{array}{rcl} 324 &=& s^2\\ \sqrt{324} &=& \sqrt{s^2}\\ 324 &=& 18 \times 18\\ s^2 &=& s \times s\\ 18 &=& s \end{array}\end{align*}

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.

\begin{align*}P = 4s\end{align*}

Such that ‘\begin{align*}P\end{align*}’ is the perimeter and ‘\begin{align*}s\end{align*}’ is the length of one side of the square.

Next, fill in the value for ‘\begin{align*}s\end{align*}’ and perform the indicated operation.

\begin{align*}\begin{array}{rcl} P &=& 4s\\ P &=& 4(18)\\ P &=& 72 \end{array}\end{align*}

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.

\begin{align*}\begin{array}{rcl} x^2 &=& 26\\ \sqrt{x^2} &=& \sqrt{26} \end{array}\end{align*}

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 2nd \begin{align*}x^2 \quad 2 \quad 6\end{align*}  enter.

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

\begin{align*}\begin{array}{rcl} \sqrt{x^2} &=& \sqrt{26}\\ x &=& 5.099019514\\ x &=& 5.1 \end{array}\end{align*}

#### Example 3

Solve the following radical equation for the variable ‘\begin{align*}m\end{align*}’.

\begin{align*}m^2 = 144\end{align*}

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

\begin{align*}\begin{array}{rcl} m^2 &=& 144\\ \sqrt{m^2} &=& \sqrt{144} \end{array}\end{align*}

Next, simplify both sides of the equation.

\begin{align*}\begin{array}{rcl} \sqrt{m^2} &=& \sqrt{144}\\ m^2 &=& m \times m\\ 144 &=& 12 \times 12\\ m &=& 12 \end{array}\end{align*}

#### Example 4

Solve the following radical equation for the variable ‘\begin{align*}a\end{align*}’.

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

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

\begin{align*}\begin{array}{rcl} \sqrt{a-8} &=& 7\\ \left(\sqrt{a-8} \right)^2 &=& (7)^2 \end{array}\end{align*}

Next, perform any indicated operations and simplify the equation.

\begin{align*}\begin{array}{rcl} \left(\sqrt{a-8} \right)^2 &=& (7)^2\\ a-8 &=& 7 \times 7\\ a-8 &=& 49 \end{array}\end{align*}

Then, add eight to both sides of the equation to solve for the variable ‘\begin{align*}a\end{align*}’.

\begin{align*}\begin{array}{rcl} a-8 &=& 49\\ a-8+8 &=& 49+8\\ a &=& 57 \end{array}\end{align*}

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

\begin{align*}\begin{array}{rcl} \sqrt{a-8} &=& 7\\ \sqrt{{\color{blue}57}-8} &=& 7 \end{array}\end{align*}

Then, perform any indicated operations and simplify the equation.

\begin{align*}\begin{array}{rcl} \sqrt{{\color{blue}57}-8} &=& 7\\ \sqrt{49} &=& 7\\ 49 &=& 7 \times 7\\ 7 &=& 7 \end{array}\end{align*}

#### Example 5

Solve the following equation for the variable.

\begin{align*}x^3+4 = 31\end{align*}

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.

\begin{align*}\begin{array}{rcl} x^3+4-4 &=& 31-4\\ x^3 &=& 27 \end{array}\end{align*}

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

\begin{align*}\begin{array}{rcl} x^3 &=& 27\\ \sqrt[3]{x^3} &=& \sqrt[3]{27} \end{array}\end{align*}

Then, perform the indicated operations.

\begin{align*}\begin{array}{rcl} \sqrt[3]{x^3} &=& \sqrt[3]{27}\\ \sqrt[3]{x^3} &=& x \times x \times x\\ \sqrt[3]{27} &=& 3 \times 3 \times 3\\ x &=& 3 \end{array}\end{align*}

### Review

Solve each equation involving radical expressions.

1. \begin{align*}x^2 = 121\end{align*}
2. \begin{align*}x^2 = 144\end{align*}
3. \begin{align*}x^2 = 64\end{align*}
4. \begin{align*}x^2 = 169\end{align*}
5. \begin{align*}x^2 = 16\end{align*}
6. \begin{align*}x^3 = 64\end{align*}
7. \begin{align*}x^3 = 27\end{align*}
8. \begin{align*}x^2+3 = 147\end{align*}
9. \begin{align*}x^2-2 = 23\end{align*}
10. \begin{align*}x^2+5 = 30\end{align*}
11. \begin{align*}x^3+4 = 68\end{align*}
12. \begin{align*}x^3+10 = 135\end{align*}
13. \begin{align*}x^2-4 = 21\end{align*}
14. \begin{align*}x^2-6 = 30\end{align*}
15. \begin{align*}x^2-12 = 37\end{align*}

### Vocabulary Language: English

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $32^4$, 32 is the base, and 4 is the exponent.

Cubed

The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = $2^3 = 2 \times 2 \times 2 = 8$.

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

Extraneous Solution

An extraneous solution is a solution of a simplified version of an original equation that, when checked in the original equation, is not actually a solution.

Fractional Power

A fractional power is an exponent in fraction form. A fractional exponent of $\frac{1}{2}$ is the same as the square root of a number. A fractional exponent of $\frac{1}{3}$ is the same as the cube root of a number.

Perfect Square

A perfect square is a number whose square root is an integer.

A quadratic equation is an equation that can be written in the form $=ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real constants and $a\ne 0$.

The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

A radical expression is an expression with numbers, operations and radicals in it.

Squared

Squared is the word used to refer to the exponent 2. For example, $5^2$ could be read as "5 squared". When a number is squared, the number is multiplied by itself.