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## Find the roots of basic equations containing roots

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Have you ever planted a garden? Take a look at this dilemma.

Mario and his brother have built a garden bed with an area of 144 square feet. If the shape of the garden is a square, then what is the length of one side of the garden? What is the perimeter of this garden?

To figure this out, you will need to understand how to solve equations that involve radicals. This Concept will teach you how to find success.

### Guidance

Do you remember exponents?

An exponent is a number that raises a base to a power. We can recognize exponents because they are little numbers next to larger numbers. The little number is the exponent and the large number is the base. The exponent tells you how many times to multiply a base by itself.

72\begin{align*}7^2\end{align*}

This means that we multiply the base of 7 by itself two times. This is how we evaluate a power.

7×7=49\begin{align*}7 \times 7 = 49\end{align*}

We can also perform an operation that is the opposite of raising a number to a power; we can find the root of a number. This is an expression that is the opposite of raising a number to a power. We call it a root or a radical.

When you see a number that looks like this, 16\begin{align*}\sqrt{16}\end{align*}, this means that we are looking for the root of the number that is inside the radical symbol.

In this case, the answer would be 4 because 4 x 4 = 16.

We can also see radicals in an equation. When we have a radical in an equation, we can solve the equation by using what we have learned about squares and square roots. Let’s take a look.

x2=100\begin{align*}x^2=100\end{align*}

To solve this equation let’s think about what we know. We know that a value times itself is going to equal 100. Up until this point we have solved equations by using the inverse operation. We can use that again here too. The variable is squared; we want to get it by itself by using an inverse operation. The inverse of squaring a number is to find the square root of the number. Let’s see what happens if we find the square root of both sides of the equation.

x2=100\begin{align*}\sqrt{x^2}= \sqrt{100}\end{align*}

The exponent 2 and the radical cancel each other out because they are inverses of each other. Let’s cross them out.

x2=100\begin{align*}\sqrt{x^2}=\sqrt{100}\end{align*}

Now are left with:

xx=100=10

Solve each equation.

#### Example A

x2=36\begin{align*}x^2=36\end{align*}

Solution:  x=6\begin{align*}x = 6\end{align*}

#### Example B

x2=81\begin{align*}x^2=81\end{align*}

Solution:  x=9\begin{align*}x = 9\end{align*}

#### Example C

x3+2=66\begin{align*}x^3+2=66\end{align*}

Solution:  x=4\begin{align*}x = 4\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

First, we have to break down what we know about the garden. We know the area.

A=144 sq.feet\begin{align*}A = 144 \ sq.feet\end{align*}

We also know that the formula for area is s2\begin{align*}s^2\end{align*}.

Now we can write an equation to figure out the first part of the problem, the length of one of the sides.

s2=144 sq.feet\begin{align*}s^2=144 \ sq.feet\end{align*}

If we take the square root of both sides, then we will be able to solve for the value of the side of the garden.

sqrts2s=144=12 feet

The length of one of the sides of the garden is 12 feet.

Next, we need to figure out the perimeter of the garden.

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

Let’s substitute in the known value for the side of the garden.

P=4(12)\begin{align*}P=4(12)\end{align*}

The perimeter of the garden is 48 square feet.

### Vocabulary

Exponent
the little number that represents a power. It tells you how many times to multiply the base by itself.
Base
the number being raised to a power. It is the large number next to an exponent.
a number inside a radical where you will need to find the root of a number.
Squared
an exponent of 2, tells you to multiply the base by itself.
Cubed
an exponent of 3, tells you to multiply the base by itself three times.
Cube Root
to find a value that when multiplied by itself three times is equal to the value inside the radical.
Perfect Square
A number that is a square of a whole number.
Perfect Cube
a number that is the cube of a whole number.
Fractional Power
an exponent in fraction form. A fractional exponent of 12\begin{align*}\frac{1}{2}\end{align*} is the same as the square root of a number. A fractional exponent of 13\begin{align*}\frac{1}{3}\end{align*} is the same as the cube root of a number.

### Guided Practice

Here is one for you to try on your own.

x2=121\begin{align*}x^2=121\end{align*}

Solution

To solve this equation let’s think about what we know. We know that a value times itself is going to equal 100. Up until this point we have solved equations by using the inverse operation. We can use that again here too. The variable is squared; we want to get it by itself by using an inverse operation. The inverse of squaring a number is to find the square root of the number. Let’s see what happens if we find the square root of both sides of the equation.

x2=121\begin{align*}\sqrt{x^2}= \sqrt{121}\end{align*}

The exponent 2 and the radical cancel each other out because they are inverses of each other. Let’s cross them out.

x2=121\begin{align*}\sqrt{x^2}=\sqrt{121}\end{align*}

Now are left with:

xx=121=11

### Practice

Directions: Solve each equation involving radical expressions.

1. x2=121\begin{align*}x^2=121\end{align*}
2. x2=144\begin{align*}x^2=144\end{align*}
3. x2=64\begin{align*}x^2=64\end{align*}
4. x2=169\begin{align*}x^2=169\end{align*}
5. x2=16\begin{align*}x^2=16\end{align*}
6. x3=64\begin{align*}x^3=64\end{align*}
7. x3=27\begin{align*}x^3=27\end{align*}
8. x2+3=147\begin{align*}x^2+3=147\end{align*}
9. x22=23\begin{align*}x^2-2=23\end{align*}
10. x2+5=30\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

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

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

Exponent

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

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

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

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

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.