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1. Real Numbers
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United States of AmericaVirginia
Grade 8

1.02 Estimate square roots

Lesson
Practice

Estimate square roots

The value of \pi and square roots of non-perfect squares are examples of irrational numbers. Recall the decimal expansion of an irrational number is a non-terminating, non-repeating decimal.

So far, we have estimated the values of irrational numbers using a calculator. However, there are ways we can approximate the values of irrational numbers without using a calculator.

Perfect square

a whole number whose square root is an integer

Square root (of a number)

any number which, when multiplied by itself, equals the given number

Exploration

Move the slider to change the area of the square. Examine how the side lengths of the square change.

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  1. Why is the side length of a square with an area of 25 units^2 equal to \sqrt{25}?

  2. What do you notice about the areas of squares with whole number side lengths?

  3. What do you notice about the side lengths of squares that have an area which is not a perfect square?

  4. What are the closest perfect square area values that are smaller and larger than an area of 30 square units?

  5. What does this tell us about the whole number side length values that \sqrt{30} lies between? What are they?

The positive and negative square root of any whole number that is not a perfect square lies between two consecutive integers.

In other words, if a \lt b \lt c

then

\sqrt{a} \lt \sqrt{b} \lt \sqrt{c}

We can use this to approximate the location of the square root of a non-perfect square on the number line by finding perfect squares that lie on either side of that number.

For example, let's look at \sqrt{75}.

First, we identify that

64 \lt 75 \lt 81

We know that 64 and 81 are perfect squares since 8^{2} = 64 and 9^{2} = 81.

This means that \sqrt{64} \lt \sqrt{75} \lt \sqrt{81}

and so 8 \lt \sqrt{75} \lt 9

So we can place \sqrt{75} on the number line between 8 and 9. But where should we put it? Closer to 8? Closer to 9? Right in the middle?

number line from 8-9. 3 dots plotted: one in the middle, the other two close to 8 and 9. all marked with questions marks.

We know that the \sqrt{75} is between the natural numbers 8 and 9, so we can find the differences between 64, 75, and 81 to get a better understanding of where the \sqrt{75} lies on the number line.

Since 75-64=11 and 81-75=6. Because 11\gt 6, we can see that 75 is closer to 81 than it is to 64. This means that \sqrt{75} is closer to 9.

This tells us that the \sqrt{75} is somewhere in this region on the numberline between 8.5 and 9.

A numberline showing major tickmarks of 8 and 9. A dot is plotted at the center of the two numbers. Another dot is plotted somewhere between the middle dot and 9, and labeled with a question mark. The line from the middle dot to 9 is highlighted in red.

If we need to get a better estimate of the value of \sqrt{75}, we can start by drawing a grid where each square represents one unit of area.

A 9x9 grid. the 8x8 part is blue and the outer part is green. 11 of the green part is marked 'x', while all the blue parts are marked with 'x'.

On this grid, we plot a square sized 8 \times 8 square grid shaded in blue to represent 64 square units. This grid is placed over top of a 9 \times 9 square grid shaded in green to represent 81 square units.

We need to mark a total of 75 squares with an 'X' starting with the 64 blue squares. We then mark 11 green squares with an 'X' to have a total of 75 squares marked with an 'X'. Each 'X' represents one unit.

Since more than half of the green squares were marked with an 'X' we know that \sqrt{75} is closer to 9 than it is to 8.

We can estimate the value of \sqrt{75} to be approximately 8 \dfrac{11}{17}, where 11 represents the number of 'X' marked green squares and 17 represents the total number of squares that would complete the 9 \times 9 grid.

We can then mark \sqrt{75} on the numberline where 8 \dfrac{11}{17} is.

A number line from 8 to 9. A dot is plotted in between, closer to 9

This estimation is fairly close to the \sqrt{75}. If we use our calculator to get an approximate value we get \sqrt{75} \approx 8.66 and 8 \dfrac{11}{17} \approx 8.65. We can see that our fractional estimate is a good approximation for the \sqrt{75}.

Examples

Example 1

Approximate \sqrt{95} to the nearest tenth without using a calculator.

a

Identify the two natural numbers the square root lies between.

Worked Solution
Create a strategy

To find the two natural numbers that \sqrt{95} lies between, we need to find the two perfect squares that are just less than and just greater than 95. Then we will calculate their square roots.

Apply the idea

First, we identify the two closest perfect squares to 95, which are 81 and 100.

We can order these numbers using inequality signs:

81 \lt 95 \lt 100

We know this order will hold true if we take the square root of each term.

This means that \sqrt{81} \lt \sqrt{95} \lt \sqrt{100} and since 81 and 100 are perfect squares, we can simplify them to get 9 \lt \sqrt{95} \lt 10

This means that \sqrt{95} is between the natural numbers 9 and 10.

b

Approximate \sqrt{95} to the nearest tenth without using a calculator.

Worked Solution
Create a strategy

We can use a grid to estimate the value of \sqrt{95}.

Apply the idea

We identified in part (a) that \sqrt{95} lies between 9 and 10.

To determine whether \sqrt{95} is closer to 9 or 10, we will look at how close 95 is to 9^2 and 10^2.

Create 9\times9 and 10\times 10 grids, which represent 9^2 and 10^2, and place the smaller grid on top of the larger one.

The length of the total grid represents \sqrt{100}=10 and the length of the blue grid represents \sqrt{81}=10.

A 9x9 blue grid on top of a 10x10 green grid. There are labels on each side showing how many squares there are.

Mark the area up to 95 squares starting with the 81 blue squares. One square is equal to one unit. We cannot make a square with whole number side lengths using 95 smaller squares, so we can use the unmarked green squares to visually approximate how much closer the \sqrt{95} is to 10 than to 9.

A 9x9 blue grid on top of a 10x10 green grid. 95 units are marked with ×

The blue squares represent the whole number of 9. The green squares represent a fraction, where the marked ones are the numerator and the total number of green squares is the denominator. There are 14 marked green squares and 19 total green squares, so the fraction part is \dfrac{14}{19}.

We then can combine the whole number, 9, and the fraction, \dfrac{14}{19}, to get our estimation for the \sqrt{95}.

\sqrt{95} \approx 9\dfrac{14}{19} or 9.7

Reflect and check

We can use a calculator to square the fraction to see how close it is to 95.

\sqrt{95}^{2} \approx \left( 9\dfrac{14}{19} \right)^{2}

95 \approx 94.806

Our fractional estimate of 9\dfrac{14}{19} for the \sqrt{95} is fairly close. It is only about 0.194 off from the exact number.

If we want to represent \sqrt{95} using a square, its side lengths would be approximately 9.7\text{ units}.

A 9.7x9.7 square grid. the 9x9 is in blue and 0.7 x 0.7 is in green.

Example 2

Given the irrational numbers \sqrt{2}, \sqrt{50}, \pi, \sqrt{29}, \sqrt{98}.

a

Plot the irrational numbers on the following number line:

012345678910
Worked Solution
Create a strategy

Estimate the values of the numbers by determining which two perfect squares each number lies between. Then plot the estimations on the same number line.

Apply the idea

Estimate the value of \sqrt{2} by comparing to the closest square numbers that are bigger and smaller. \begin{aligned} \sqrt{1} \lt &\sqrt{2} \lt \sqrt{4}\\\\ 1 \lt &\sqrt{2} \lt 2 \end{aligned}

The value of \sqrt{2} is closer to \sqrt{1} than \sqrt{4} since 2-1=1 and 4-2=2.

So, we can place \sqrt{2} between 1 and 2 but closer to 1 on the number line.

012345678910

Estimate the value of \sqrt{50} by comparing to the closest square numbers that are bigger and smaller. \begin{aligned} \sqrt{49} \lt &\sqrt{50} \lt \sqrt{64}\\\\ 7 \lt &\sqrt{50} \lt 8 \end{aligned}

The value of \sqrt {50} is closer to \sqrt {49} than \sqrt {64} since 50-49=1 and 64-50=14. So, we can have \sqrt{50} plotted between 7 and 8 but closer to 7.

012345678910

Estimate the value of \sqrt{29} by comparing to the closest square numbers that are bigger and smaller. \begin{aligned} \sqrt{25} \lt &\sqrt{29} \lt \sqrt{36}\\\\ 5 \lt &\sqrt{29} \lt 6 \end{aligned}

The value of \sqrt {29} is closer to \sqrt {25} than \sqrt {36} since 29-25=4 and 36-29=7.

So, we can place \sqrt{29} between 5 and 6 but closer to 5.

012345678910

Estimate the value of \sqrt{98} by comparing to the closest square numbers that are bigger and smaller. \begin{aligned} \sqrt{89} \lt &\sqrt{98} \lt \sqrt{100}\\\\ 9 \lt &\sqrt{98} \lt 10 \end{aligned}

The value of \sqrt {98} is closer to \sqrt {100} than \sqrt {81} since 100-98=2 and 98-89=17.

So, we can place \sqrt{98} between 9 and 10 but closer to 10.

012345678910

Recall that \pi estimated to two decimals is 3.14.

Plotting the final point we get the number line:

012345678910
b

Arrange the irrational numbers from largest to smallest.

Worked Solution
Create a strategy

In part a we plotted all the numbers on a number line, we can use the number line to order them from largest to smallest.

Apply the idea

Based on the positions of the numbers on the number line, going right to left, the order is:

\sqrt{98}, \sqrt{50},\sqrt{29},\pi,\sqrt{2}

Reflect and check

We can also determine the order of numbers from largest to smallest by squaring each of our numbers confirming they are in the same order we found. We will estimate that \pi = 3.14.

\sqrt{98}^{2}, \sqrt{50}^{2},\sqrt{29}^{2},3.14^{2},\sqrt{2}^{2}

98,50,29,9.8596,2

The numbers are still in order from largest to smallest.

Idea summary

We can estimate the values of irrational numbers and represent them on the number line.

To estimate square roots we can follow these steps:

  • Determine the closest perfect squares that are bigger and smaller than the number underneath the root symbol.

  • Find the square roots of the perfect squares to find what two whole numbers the given number lies between.

To determine which natural number the given square root lies closet to, we can find the differences between the perfect squares and the number underneath the square root symbol.

Outcomes

8.NS.1

The student will compare and order real numbers and determine the relationships between real numbers.

8.NS.1a

Estimate and identify the two consecutive natural numbers between which the positive square root of a given number lies and justify which natural number is the better approximation. Numbers are limited to natural numbers from 1 to 400.

8.NS.1b

Use rational approximations (to the nearest hundredth) of irrational numbers to compare, order, and locate values on a number line. Radicals may include both positive and negative square roots of values from 0 to 400 yielding an irrational number.

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