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# Simplification of Radical Expressions

## Evaluate and estimate numerical square and cube roots

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License: CC BY-NC 3.0

Mark, a huge baseball fan, asked his sister Sara, “Did you know that the distance between any two bases on a baseball diamond is equal to the square root of 8100?”

How can Sara learn or estimate the square root of 8100?

In this concept, you will learn to evaluate radical expressions.

An exponent is the small number to the upper right of a base that tells you how many times to multiply the base times itself. For example 72=7×7=49\begin{align*}7^2 = 7 \times 7 = 49\end{align*}. The base of 7 is raised to the power of 2 which means to multiply 7 times itself twice. The answer is 49.

Finding the square root of a number is the inverse operation of squaring. The square root of a number ‘n\begin{align*}n\end{align*},’ denoted by n\begin{align*}\sqrt{n}\end{align*}, is a positive number whose square is n\begin{align*}n\end{align*}. The n\begin{align*}\sqrt{n}\end{align*} is called a radical. A radical is an expression consisting of a radical sign \begin{align*}\sqrt{}\end{align*} , a radicand which is the number or term under the radical sign and an index which is the small number outside the radical sign which indicates what root to find. The index for ‘square root’ is two but it is never written in the expression - it is understood since the radical sign with no index is referred to as the square root.

The following is an example of a radical.

radical signindex643radicand\begin{align*}\begin{array}{rcl} && \qquad \ \ \ \ \ \downarrow \text{radical sign}\\ && {\color{red}\text{index} \rightarrow} \sqrt[{\color{red}3}]{{\color{blue}64}} {\color{blue}\leftarrow \text{radicand}} \end{array}\end{align*}

When a number is written under a radical sign, you are finding the square root of that number. For the radical 49\begin{align*}\sqrt{49}\end{align*} you are finding the number times itself twice that equals 49. The answer is 7. Forty-nine is called a perfect square because it is a number that can be written as a power with an integer base and the exponent 2.

A number raised to the power of 3 is said to be cubed. For example 43=4×4×4=64\begin{align*} 4^3 = 4 \times 4 \times 4 = 64\end{align*}. The base of 4 has been multiplied by itself three times. The inverse operation would involve finding the cube root of 64

The cube root of 64 is the number times itself three times that equals 64. The expression can be written as 643=3\begin{align*} \sqrt[3]{64} =3\end{align*} . Sixty-four is called a perfect cube because it is a number that can be written as a power with an integer base and the exponent 3.

Not all numbers are perfect squares or perfect cubes. The values for these numbers can either be estimated or found using the TI calculator. If you were asked to find the square root of thirty then you could estimate the answer.

30\begin{align*}\sqrt{30}\end{align*} is between the two perfect squares 25=52 and 36=62\begin{align*} 25 = 5^2 \text{ and } 36 = 6 ^ 2\end{align*} . Since 30 is approximately half-way between 25 and 36, then an estimate of 5.5 would be a reasonable answer for the square root of 30.

You could also determine the value of 30\begin{align*}\sqrt{30}\end{align*} using the TI calculator.

Using the TI calculator press the keys in the order shown below.[Figure2]

License: CC BY-NC 3.0

The estimate of 5.5 is the same as the calculator answer if it were rounded to the nearest tenth.

Estimate the following:

353\begin{align*}\sqrt[3]{35}\end{align*}

Thirty-five is not a perfect cube since there is no integer when multiplied by itself 3 times equal to 35.

The value 35 is between the perfect cube 27=3×3×3=33\begin{align*}27= 3 \times 3 \times 3 = 3 ^3 \end{align*} and 64=4×4×4=43\begin{align*}64= 4 \times 4 \times 4 = 4 ^ 3\end{align*}. The number 35 is closer to 27 than it is to 64. Therefore an estimate for the cube root of 35 is 3.3.

### Examples

#### Example 1

Earlier, you were given a problem about Mark and the baseball diamond. His sister wants to figure out the square root of 8100.

First, rewrite the square root of 8100 as a radical.

8100\begin{align*}\sqrt{8100}\end{align*}

Next, write the square root of 8100 as 81×100\begin{align*}\sqrt{81} \times \sqrt{100} \end{align*}

Next, write the square of 81.

81=9×9=92\begin{align*}81= 9 \times 9 = 9^2 \end{align*} The square of 81 is 9.

Next, write the square of 100.

100=10×10=102\begin{align*}100 = 10 \times 10 = 10^2\end{align*} The square of 100 is 10.

Then, multiply the two squares.

9×10=90\begin{align*}9 \times 10 = 90\end{align*}

The answer is 90.

The distance between any two bases is 90 feet.

#### Example 2

Estimate the square root of the following by listing the two squares that the root falls between:

74\begin{align*}\sqrt{74}\end{align*}

First, figure out the perfect square that is closest to but less than 74.

64=82\begin{align*}64 = 8 ^2\end{align*}

Next, figure out the perfect square that is closest to but greater than 74.

81=92\begin{align*}81 = 9^2\end{align*}

The answer is 8 and 9.

74\begin{align*}\sqrt{74}\end{align*} is between 8 and 9.

#### Example 3

Using the TI calculator, calculate the following square root to the nearest hundredth.

115\begin{align*}\sqrt{115}\end{align*}

First, press the buttons 2nd x2\begin{align*} 2 ^{nd} \ x^2 \end{align*} 1 1 5 enter

Next, look at the answer displayed on the screen of the calculator.

10.72380529\begin{align*}10.72380529\end{align*}

Then, round the answer to two places after the decimal point.

10.73\begin{align*}10.73\end{align*}

The answer is 11510.73\begin{align*} \sqrt{115} \approx 10.73\end{align*}

#### Example 4

Without using technology, which of the following numbers are perfect squares? Justify your answer.

196   120   36\begin{align*}196 \ \ \ 120 \ \ \ 36\end{align*}

Create a factor tree for each of the given numbers.

First, 196 is an even number so divide it by two.

1962=98\begin{align*}\frac{196}{2} = 98 \end{align*}

Next, 98 is an even number so divide it by two.

982=49\begin{align*}\frac{98}{2} = 49 \end{align*}

Next, write down the numbers that will multiply to give 49.

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

Next, list all the factors used to simplify 196.

2,2,7,7\begin{align*}2,2,7,7\end{align*}

Next, make two groups of same factors.

2×7 and 2×7\begin{align*}2 \times 7 \text{ and }2 \times 7\end{align*}

Then, multiply each group of factors.

14 and 14\begin{align*}14 \ \text{and} \ 14\end{align*}
The answer is 14.

196=14\begin{align*} \sqrt{196} = 14\end{align*}

First, 120 is an even number so divide it by two.

1202=60\begin{align*}\frac{120}{2} = 60\end{align*}

Next, 60 is an even number so divide it by two.

602=30\begin{align*}\frac{60}{2} = 30\end{align*}

Next, 30 is an even number so divide it by two.

302=15\begin{align*}\frac{30}{2} = 15\end{align*}

Next, write down the numbers that multiply to give 15.

3×5\begin{align*}3 \times 5\end{align*}

Next, list all the factors used to simplify 120.

2,2,2,3,5\begin{align*}2, 2, 2, 3, 5\end{align*}

Then, make two groups of same factors.

There are three 2’s, one 3, and one 5. Two groups of same factors cannot be made from these numbers.

120 is not a perfect square.

First, 36 is an even number so divide it by two.

362=18\begin{align*}\frac{36}{2} = 18\end{align*}

Next, 18 is an even number so divide it by two.

182=9\begin{align*}\frac{18}{2} = 9\end{align*}

Next, write down the numbers that multiply to give 9.

3×3\begin{align*}3 \times 3\end{align*}

Next, list all the factors used to simplify 36.

2,2,3,3\begin{align*}2, 2, 3, 3\end{align*}

Next, make two groups of same factors.

2×3 and 2×3\begin{align*}2 \times 3 \text{ and } 2 \times 3\end{align*}

Then, multiply each group of factors.

6 and 6\begin{align*}6 \ \text{and} \ 6\end{align*}

The answer is 6.

36=6\begin{align*}\sqrt{36} = 6\end{align*}

### Review

Evaluate each radical expression without using technology.

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*}

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

11. 12\begin{align*}\sqrt{12}\end{align*}

12. 15\begin{align*} \sqrt{15}\end{align*}

13. 20\begin{align*}\sqrt{20}\end{align*}

14. 22\begin{align*} \sqrt{22}\end{align*}

15. 31\begin{align*}\sqrt{31}\end{align*}

16. 90\begin{align*}\sqrt{90}\end{align*}

17. 99\begin{align*} \sqrt{99}\end{align*}

To see the Review answers, open this PDF file and look for section 7.1.

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### 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.

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

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

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

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