---
title: "3.1.9 Finding Approximations of Square Roots"
layout: ../../../../layouts/LessonLayout.astro
---
import { Question } from "../../../../components/Question";
# 3.1.9 Finding Approximations of Square Roots
Seeing a problem like approximating $\sqrt{1234567}$ is very common in the middle of the test. The basic trick is you want to “take out” factors of 100 under the radical. Let’s look at the above example after noticing that we can roughly approximate (within the margin of error) $\sqrt{1234567} \approx \sqrt{1230000}$. Now:
$$
\sqrt{1230000} = \sqrt{123 \cdot 100 \cdot 100} = 10 \cdot 10 \sqrt{123}
$$
Now we are left with a much simpler approximation of the $100 \cdot \sqrt{123} \approx 100 \cdot 11 = 1100$.
You can follow the same procedure for cubed roots as well, only you need to find factors of 1000 under the radical to take out. Let’s look at the example of $\sqrt[3]{1795953}$ after making the early approximation of $\sqrt[3]{1795953} \approx \sqrt[3]{1795000}$
$$
\sqrt[3]{1795000} = \sqrt[3]{1795 \cdot 1000} = 10 \cdot \sqrt[3]{1795}
$$
Well we should have memorized that $12^3 = 1728$ so we can form a rough approximation:
$$
10 \cdot \sqrt[3]{1795} = 10 \cdot 12.1 = 121
$$
So the trick is if you are approximating the $n$-th root of some number, you “factor out” sets of the $n$-digits and then approximate a much smaller value, then move the decimal place over accordingly.
Now in some instances you are asked to find the exact value of the cubed root. For example: $\sqrt[3]{830584}$. Now the procedure would be as followed:
1. Figure out how many digits you are going to have by noticing how many three-digit “sets” there are. Most will only be two digit numbers, however this is not guaranteed.
2. To find out the units digit, look at the units digit of the number given and think about what number cubed would give that result.
3. After that, you want to disregard the last three digits, and look at the remaining number and find out what number cubed is the first integer less than that value.
So to use the procedure give above for the problem of $\sqrt[3]{830584}$:
1. Well you have two, three-digit “sets” (the sets being 584 and 830). This means that we are looking for a two-digit number in our answer.
2. The last digit is 4, so what number cubed ends in a 4? The answer is that $4^3 = 64$ so the last digit of the answer is 4.
3. Now we disregard the first set of three (584) and look at the remaining numbers (830). So what number cubed is less than 830. Well we know $10^3 = 1000$ and $9^3 = 729$ so 9 is the largest integer so that when cubed is less than 830. So that is the tens digit.
4. The answer is 94.
The following are problems so that you can practice this procedure of finding approximate and exact values of square and cubed roots.
### Problem Set 3.1.9
<div className="not-prose grid grid-cols-2 md:grid-cols-3 gap-2 my-4">
<Question type="math_range" id="approx-square-roots-1"
question="$\sqrt{15376} =$"
answer="[117.80, 130.20]"
client:load/>
<Question type="math" id="approx-square-roots-2"
question="$\sqrt[3]{830584} =$"
answer="94"
client:load/>
<Question type="math_range" id="approx-square-roots-3"
question="$\sqrt{23456} =$"
answer="[145.35, 160.65]"
client:load/>
<Question type="math_range" id="approx-square-roots-4"
question="$\sqrt{32905} =$"
answer="[171.95, 190.05]"
client:load/>
<Question type="math_range" id="approx-square-roots-5"
question="$\sqrt{6543210} =$"
answer="[2430.10, 2685.90]"
client:load/>
<Question type="math" id="approx-square-roots-6"
question="$\sqrt[3]{658503} =$"
answer="87"
client:load/>
<Question type="math_range" id="approx-square-roots-7"
question="$\sqrt{6213457} =$"
answer="[2368.35, 2617.65]"
client:load/>
<Question type="math_range" id="approx-square-roots-8"
question="$\sqrt{173468} =$"
answer="[395.20, 436.80]"
client:load/>
<Question type="math_range" id="approx-square-roots-9"
question="$\sqrt{6420135} =$"
answer="[2407.30, 2660.70]"
client:load/>
<Question type="math_range" id="approx-square-roots-10"
question="$\sqrt{872143} =$"
answer="[887.30, 980.70]"
client:load/>
<Question type="math_range" id="approx-square-roots-11"
question="$\sqrt{272727} =$"
answer="[495.90, 548.10]"
client:load/>
<Question type="math_range" id="approx-square-roots-12"
question="$\sqrt{38527} =$"
answer="[186.20, 205.80]"
client:load/>
<Question type="math_range" id="approx-square-roots-13"
question="$\sqrt{32323} =$"
answer="[171.00, 189.00]"
client:load/>
<Question type="math_range" id="approx-square-roots-14"
question="$\sqrt{18220} =$"
answer="[128.25, 141.75]"
client:load/>
<Question type="math_range" id="approx-square-roots-15"
question="$\sqrt{25252} =$"
answer="[151.05, 166.95]"
client:load/>
<Question type="math_range" id="approx-square-roots-16"
question="$\sqrt{265278} =$"
answer="[489.25, 540.75]"
client:load/>
<Question type="math_range" id="approx-square-roots-17"
question="$\sqrt{81818} =$"
answer="[271.70, 300.30]"
client:load/>
<Question type="math_range" id="approx-square-roots-18"
question="$\sqrt{262626} =$"
answer="[486.40, 537.60]"
client:load/>
<Question type="math_range" id="approx-square-roots-19"
question="$\sqrt{765432} =$"
answer="[831.25, 918.75]"
client:load/>
<Question type="math_range" id="approx-square-roots-20"
question="$\sqrt{80808} =$"
answer="[269.80, 298.20]"
client:load/>
<Question type="math_range" id="approx-square-roots-21"
question="$\sqrt{97531} =$"
answer="[296.40, 327.60]"
client:load/>
<Question type="math_range" id="approx-square-roots-22"
question="$\sqrt{86420} =$"
answer="[279.30, 308.70]"
client:load/>
<Question type="math_range" id="approx-square-roots-23"
question="$\sqrt{8844} \times \sqrt{6633} =$"
answer="[7276.05, 8041.95]"
client:load/>
<Question type="math_range" id="approx-square-roots-24"
question="$\sqrt[3]{217777} \times \sqrt{3777} \times 57 =$"
answer="[199813.50, 220846.50]"
client:load/>
<Question type="math_range" id="approx-square-roots-25"
question="$\sqrt[3]{26789} \times \sqrt{911} \times 31 =$"
answer="[26600.00, 29400.00]"
client:load/>
<Question type="math_range" id="approx-square-roots-26"
question="$\sqrt[3]{215346} \times \sqrt{3690} \times 57 =$"
answer="[196650.00, 217350.00]"
client:load/>
<Question type="math_range" id="approx-square-roots-27"
question="$\sqrt[3]{2006} \times 6002 =$"
answer="[71820.00, 79380.00]"
client:load/>
<Question type="math_range" id="approx-square-roots-28"
question="$\sqrt[3]{63489} \times \sqrt{1611} \times 41 =$"
answer="[62320.00, 68880.00]"
client:load/>
<Question type="math_range" id="approx-square-roots-29"
question="$\sqrt[4]{14643} \times \sqrt[3]{1329} \times \sqrt{120} =$"
answer="[1244.50, 1375.50]"
client:load/>
</div>