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## Evaluate and estimate numerical square and cube roots

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Have you ever watched a baseball game? Have you ever seen a runner run so fast that he passed the intended base? Take a look at what Miguel saw.

Miguel was watching a game on Saturday night. During the fourth inning, the runner ran from second base to third base. He was running to fast that he ran past third base and had to come back to reach the base.

The runner ran 90+16\begin{align*}90 + \sqrt{16}\end{align*}.

How far did the runner run?

This situation has been described using a radical expression. You will learn how to evaluate radical expressions in this Concept.

### Guidance

Sometimes, we can have an expression with a radical in it.

A radical is the name of the sign that tells us that we are looking for a square root. We can call this “a radical.” Here is a radical symbol.

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

Here we would be looking for the square root of y\begin{align*}y\end{align*}.

Remember than an expressions is a number sentence that contains numbers, operations and now radicals. Just as we can have expressions without radicals, we can have expressions with them too.

Let's look at one.

24+7\begin{align*}2 \cdot \sqrt{4} + 7\end{align*}

Here we have two times the square root of four plus seven.

That’s a great question! We can evaluate this expression by using the order of operations.

P parentheses

E exponents (square roots too)

MD multiplication/division in order from left to right

AS addition/subtraction in order from left to right

According to the order of operations, we evaluate the square root of 4 first.

4=2\begin{align*}\sqrt{4} = 2\end{align*}

Next, we substitute that value into the expression.

22+7\begin{align*}2 \cdot 2 + 7\end{align*}

Next, we complete multiplication/division in order from left to right.

2×2=4\begin{align*}2 \times 2 = 4\end{align*}

Substitute that given value.

4+7=11\begin{align*}4 + 7 = 11\end{align*}

Let’s look at another one.

4163\begin{align*}\sqrt{4} \cdot \sqrt{16} - 3\end{align*}

Here we have two radicals in the expression. We can work the same way, by using the order of operations.

416=2=4

Substitute these values into the expression.

243\begin{align*}2 \cdot 4 - 3\end{align*}

Next, we complete multiplication/division in order from left to right.

2×4=8\begin{align*}2 \times 4 = 8\end{align*}

Finally, we complete the addition/subtraction in order from left to right.

83=5\begin{align*}8 - 3 = 5\end{align*}

#### Example A

6+949+5\begin{align*}6 + \sqrt{9} - \sqrt{49} + 5\end{align*}

Solution: 7\begin{align*}7\end{align*}

#### Example B

64÷4+13\begin{align*}\sqrt{64} \div \sqrt{4} + 13\end{align*}

Solution: 17\begin{align*}17\end{align*}

#### Example C

6(7)+1213\begin{align*}6(7) + \sqrt{121} - 3\end{align*}

Solution: 50\begin{align*}50\end{align*}

Here is the original problem once again.

Miguel was watching a game on Saturday night. During the fourth inning, the runner ran from second base to third base. He was running to fast that he ran past third base and had to come back to reach the base.

The runner ran 90+16\begin{align*}90 + \sqrt{16}\end{align*}.

How far did the runner run?

To figure this out, we have to evaluate the radical expression.

The distance from one base to another is 90 feet.

The extra distance is the 16\begin{align*}\sqrt{16}\end{align*}. Let's evaluate that part first.

16=4\begin{align*}\sqrt{16} = 4\end{align*}

Now let's add that into the original expression.

90+4=94\begin{align*}90 + 4 = 94\end{align*}

The runner ran 94 feet.

### Vocabulary

Square Number
a number of units which makes a perfect square.
Square root
a number that when multiplied by itself equals the square of the number.
Perfect Square
square roots that are whole numbers.
the symbol that lets us know that we are looking for a square root.
an expression with numbers, operations and radicals in it.

### Guided Practice

Here is one for you to try on your own.

8(7)+1449\begin{align*}8(7) + \sqrt{144} - 9\end{align*}

First, let's evaluate the square root.

144=12\begin{align*}\sqrt{144} = 12\end{align*}

Now we can substitute 12 back into the original expression and solve using the order of operations.

8(7)+129\begin{align*}8(7) + 12 - 9\end{align*}

56+129\begin{align*}56 + 12 - 9\end{align*}

689=59\begin{align*}68 - 9 = 59\end{align*}

Our answer is 59\begin{align*}59\end{align*}.

### Practice

1. 2+9+152\begin{align*}2 + \sqrt{9} + 15-2\end{align*}

2. 34+169\begin{align*}3 \cdot 4 + \sqrt{169}\end{align*}

3. 1625+36\begin{align*}\sqrt{16} \cdot \sqrt{25} + \sqrt{36}\end{align*}

4. 8112+12\begin{align*}\sqrt{81} \cdot 12 + 12\end{align*}

5. 36+4716\begin{align*}\sqrt{36} + \sqrt{47} - \sqrt{16}\end{align*}

6. 6+36+252\begin{align*}6 + \sqrt{36} + 25-2\end{align*}

7. 4(5)+92\begin{align*}4(5) + \sqrt{9}-2\end{align*}

8. 15+16+5\begin{align*}15 + \sqrt{16} + 5\end{align*}

9. 3(2)+25+10\begin{align*}3(2) + \sqrt{25} + 10\end{align*}

10. 4(7)+4912\begin{align*}4(7) + \sqrt{49}-12\end{align*}

11. 2(4)+98\begin{align*}2(4) + \sqrt{9}-8\end{align*}

12. 3(7)+25+21\begin{align*}3(7) + \sqrt{25} + 21\end{align*}

13. 8(3)36+152\begin{align*}8(3) - \sqrt{36} + 15-2\end{align*}

14. 19+14422\begin{align*}19 + \sqrt{144}-22\end{align*}

15. 3(4)+64+25\begin{align*}3(4) + \sqrt{64} + \sqrt{25}\end{align*}

### Vocabulary Language: English

Perfect Square

Perfect Square

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

The $\sqrt{}$, or square root, sign.

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.
Square Number

Square Number

A square number or perfect square is a whole number that is the square of another integer. For example, 36 is a square number because $6 \cdot 6 = 36$.
Variable Expression

Variable Expression

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