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# Simplification of Radical Expressions

## Evaluate and estimate numerical square and cube roots

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Have you ever played baseball or thought about the measurements of a baseball diamond? Take a look at this dilemma.

"Did you know that the distance between any two bases on a baseball diamond is the same as the square root of 8100?" Mark asked his sister Sara one afternoon.

Sara looked at Mark.

"Are you sure?" Sara asked trying to figure out the problem in her head.

Mark simply smiled.

Is Mark correct?

This Concept is all about square roots and perfect squares. You will know how to figure out this dilemma by the end of the Concept.

### 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.

Take a look at this value.

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*}

This is the answer.

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.

Now let’s look at how we can work with roots and radicals.

Think about the value from above.

72=7×7=49\begin{align*}7^2 = 7 \times 7 = 49\end{align*}

If we use verbal language to explain this, we can say that seven squared is equal to 49. When the exponent is a 2, we can say that the number is squared because it is multiplied by itself.

We can work in the opposite of squaring, and find the square root of a number.

49\begin{align*}\sqrt {49}\end{align*}

When we see a number inside the radical symbol, we are looking to figure out the square root of that number. In other words, what times itself two times is equal to the value inside the radical symbol.

The answer is 7 because 7 squared is equal to 49.

We can also cube a number. When a number is cubed, the exponent is a 3. This means that we multiply the base by itself three times.

23\begin{align*}2^3\end{align*}

This means that we multiply the base two by itself three times.

2×2×2=8\begin{align*}2 \times 2 \times 2 = 8\end{align*}

We can also find the cube root of a number.

83\begin{align*}\sqrt[3]{8}\end{align*}

When looking for a cube root, we are looking for a value that we can multiply by itself three times.

The cube root of 8 is 2.

We can determine if a value is a perfect square or a perfect cube.

Is 64 a perfect square? Is it a perfect cube?

To figure this out, we need to look and see if there is one number that we can multiply by itself to equal 64. This means that we are looking for the square root of 64.

Let’s evaluate this radical expression.

64=8\begin{align*}\sqrt {64}=8\end{align*}

Since 82=64\begin{align*}8^2 = 64\end{align*}, 64 is a perfect square.

Now is it a perfect cube? To figure this out, we need to see if there is a value that when cubed is equal to 64.

643=4\begin{align*}\sqrt[3]{64}=4\end{align*}

Since 4×4×4=64\begin{align*}4 \times 4 \times 4 = 64\end{align*}, 64 is a perfect cube.

64 is a perfect square and a perfect cube.

Write down the definition of a radical expression, a square root and a cube root in your notebook. Also be sure that you understand how to figure out if a number is a perfect square or a perfect cube.

Roots are pretty easy to find when the numbers are perfect squares or perfect cubes. When you can’t find a root that easily squares or cubes, you will need to use a different method. If you just need to estimate a square root, you can identify which two numbers a root would be between.

If you need more information, you can refer to a table that shows square roots, or use a calculator to find an exact decimal point.

That is exactly what we are going to look at next.

Evaluate 30\begin{align*}\sqrt{30}\end{align*}

We know by looking at 30 that it is not a perfect square. Therefore, we will need to approximate the square root. We can do this by figuring out the two values that the square root will be between.

Which values squared will equal a number close to 30?

5262=25=36

We can say that the square root of 30 is between 5 and 6. Since 25 and 36 are almost the same number of units apart, we can say that an approximate answer for the square root of 30 is 5.5.

Sometimes, you will want an answer that is exact. An approximate answer will not work, when this happens, you will need to use a calculator or a table. There are tables that will tell you the exact square root of a number.

Use your calculator to find the value of 42\begin{align*}\sqrt{42}\end{align*}.

Type the square root symbol and 42 to find the square root of 42 on your calculator.

The result shown is 6.48074069840786. You can round this to the nearest hundredth and record the value as 6.48.

#### Example A

Evaluate 62\begin{align*}\sqrt {62}\end{align*}

Solution: The answer will be between 7 and 8.

#### Example B

Evaluate 11\begin{align*}\sqrt {11}\end{align*}

Solution: The answer will be between 3 and 4.

#### Example C

Is 16 a perfect square?

Solution: Yes, because 4×4=16\begin{align*}4 \times 4 = 16\end{align*}.

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

To figure this out, we have to take the square root of 8100. If you look back at the diagram from the beginning of the Concept, you will see that the distance between the bases is 90 feet.

We are going to prove that the square root of 8100 is equal to 90 feet. If this is accurate, then Mark will be correct.

8100\begin{align*}\sqrt {8100}\end{align*} = 90×90\begin{align*}90 \times 90\end{align*} or 902\begin{align*}90^2\end{align*}

Mark is correct!

### 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.

### Guided Practice

Here is one for you to try on your own.

Is 40 a perfect square?

Solution

To figure this out, we need to figure out the square root of 40.

Let’s evaluate this radical expression.

40\begin{align*}\sqrt {40}\end{align*}

There is not a number that when squared is equal to 40 without any remainders, so 40 is not a perfect square.

### Practice

Directions: Evaluate each radical expression.

1. 16\begin{align*}\sqrt{16}\end{align*}
2. 25\begin{align*}\sqrt{25}\end{align*}
3. 81\begin{align*}\sqrt{81}\end{align*}
4. 121\begin{align*}\sqrt{121}\end{align*}
5. 36\begin{align*}\sqrt{36}\end{align*}
6. 169\begin{align*}\sqrt{169}\end{align*}
7. 1253\begin{align*}\sqrt[3]{125}\end{align*}
8. 643\begin{align*}\sqrt[3]{64}\end{align*}
9. 273\begin{align*}\sqrt[3]{27}\end{align*}
10. 144\begin{align*}\sqrt{144}\end{align*}

Directions: Approximate each square root by listing the two values that the square root can be found between.

1. 12\begin{align*}\sqrt{12}\end{align*}
2. 15\begin{align*}\sqrt{15}\end{align*}
3. 20\begin{align*}\sqrt{20}\end{align*}
4. 22\begin{align*}\sqrt{22}\end{align*}
5. 31\begin{align*}\sqrt{31}\end{align*}
6. 90\begin{align*}\sqrt{90}\end{align*}
7. 99\begin{align*}\sqrt{99}\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.
Perfect Square

Perfect Square

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

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

Rationalize the denominator

To rationalize the denominator means to rewrite the fraction so that the denominator no longer contains a radical.
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.
Variable Expression

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.