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Evaluate and estimate numerical square and cube roots

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Marco bought a square picture frame that measured 144 square inches. The wooden trim was \begin{align*}1.5^{{\prime}{\prime}}\end{align*} wide all the way around. How can Marco determine what size picture will fit the frame?

In this concept, you will learn how to simplify expressions that contain radicals.

The square root symbol, \begin{align*}\sqrt{x}\end{align*}, is also called a radical.

You may already know that an expression is a mathematical sentence that contains numbers and operations. Sometimes radicals are included in a sentence.

Let’s look at an example.

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

This sentence says “two times the square root of four plus seven.”

First, remember the Order of Operations.

Parentheses

Exponents (including square roots)

Multiplication and Division in order from left to right

Addition and Subtraction in order from left to right

Next, according to those rules, evaluate the square root of 4.

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

Next, substitute that value into the expression.

\begin{align*}2 \times 2 + 7\end{align*}

Then, complete the multiplication.

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

And substitute that given value.

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

Let’s look at another example.

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

\begin{align*}\begin{array}{rcl} \sqrt{4} &=& 2 \\ \sqrt{16} &=& 4 \end{array}\end{align*}

Next, substitute these values into the expression.

\begin{align*}2 \times4-3\end{align*}

Next, complete the multiplication.

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

Then, complete the subtraction.

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

Examples

Example 1

Earlier, you were given a problem about Marco and his 144-square-inch picture frame with the wooden trim.

He needs to figure out what size picture he can put in.

First, you know that the length of each side can be determined by taking a square root.

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

Next, you know how much space the frame uses on each side.

\begin{align*}1.5 + 1.5 =\end{align*}

Then, subtract the value of the wooden frame from the length of the side to get the side dimension for a photo.

\begin{align*}\sqrt{144} - 3 =\end{align*}

\begin{align*}12-3=9\end{align*}

The answer is 9. Marco can use a 9 inch by 9 inch photo in his new frame.

Example 2

Evaluate the expression.

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

First, solve the radicals for the square roots.

\begin{align*}\begin{array}{rcl} \sqrt{9} &=& 3 \\ \sqrt{49} &=& 7 \end{array}\end{align*}

Next, substitute these values in the expression.

\begin{align*}6+3-7+5\end{align*}

Example 3

\begin{align*}6 + \sqrt{81} -10\end{align*}

\begin{align*}\sqrt{81} = 9\end{align*}

Next, substitute that value in the expression.

\begin{align*}6 + 9 - 10\end{align*}

Example 4

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

First, determine the square roots.

\begin{align*}\begin{array}{rcl} \sqrt{64} &=& 8 \\ \sqrt{4} &=& 2 \end{array}\end{align*}

Next, substitute those values in the expression.

\begin{align*}8 \div 2+13 \end{align*}

Then divide and substitute.

\begin{align*}4 + 13 \end{align*}

Example 5

Tiffany needs to trim a square piece of wood for her shop class. The board is 25 square inches. If Tiffany cuts off 1 inch all the way around, write the mathematical expression for the side dimensions of the new piece of wood. Then solve for the dimension.

First, draw a picture.

Next, you know that the original board is square and covers 25 square inches, so each side of the original board is

\begin{align*}\text{old } s = \sqrt{25}\end{align*}

Then, you know that Tiffany will cut off one inch all the way around for a total of two inches subtracted from each side. The mathematical expression for the new side is:

\begin{align*}\text{new side} = \sqrt{25} - 2\end{align*}

Take the square root, then subtract.

\begin{align*}\begin{array}{rcl} \text{new side} &=& 5-2 \\ \text{new side} &=& 3 \ in. \end{array}\end{align*}

The answer is 3 inches. The new side is 3 inches.

Review

1. \begin{align*}2 + \sqrt{9} + 15 -2\end{align*}
2. \begin{align*}3 \cdot 4 + \sqrt{169}\end{align*}
3. \begin{align*}\sqrt{16} \cdot \sqrt{25} + \sqrt{36}\end{align*}
4. \begin{align*}\sqrt{81} \cdot 12 + 12\end{align*}
5. \begin{align*}\sqrt{36} + \sqrt{49} - \sqrt{16}\end{align*}
6. \begin{align*}6 + \sqrt{36} + 25 -2\end{align*}
7. \begin{align*}4 (5) + \sqrt{9} -2\end{align*}
8. \begin{align*}15 + \sqrt{16} + 5\end{align*}
9. \begin{align*}3 (2) + \sqrt{25} + 10\end{align*}
10. \begin{align*}4(7) + \sqrt{49} - 12\end{align*}
11. \begin{align*}2(4)+\sqrt{9}-8\end{align*}
12. \begin{align*}3 (7) + \sqrt{25} + 21\end{align*}
13. \begin{align*}8 (3) - \sqrt{36} + 15 -2\end{align*}
14. \begin{align*}19 + \sqrt{144} - 22\end{align*}
15. \begin{align*}3(4) + \sqrt{64} + \sqrt{25}\end{align*}

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Perfect Square A perfect square is a number whose square root is an integer.
Radical The $\sqrt{}$, or square root, sign.
Rationalize the denominator To rationalize the denominator means to rewrite the fraction so that the denominator no longer contains a radical.
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 A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.