1/3c increased by the square root of d square root of d: sqrt(d) 1/3c increased by the square root of d [B]1/3c + sqrt(d)[/B]

1/n^2 = 3/192 Cross multiply: 192 * 1 = 3 * n^2 3n^2 = 192 Divide each side by 3: 3n^2/3 = 192/3 Cancel the 3's on the left side: n^2 = 64 Take the square root of both sides: n = [B]8 or -8[/B]

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey? [U]Set up the relationship of still water speed and downstream speed[/U] Speed down stream = Speed in still water + speed of the current Speed down stream = x+2 Therefore: Speed upstream =x - 2 Since distance = rate * time, we rearrange to get time = Distance/rate: 15/(x+ 2) + 15 /(x- 2) = 3 Multiply each side by 1/3 and we get: 5/(x + 2) + 5/(x - 2) = 1 Using a common denominator of (x + 2)(x - 2), we get: 5(x - 2)/(x + 2)(x - 2) + 5(x + 2)/(x + 2)(x - 2) (5x - 10 + 5x + 10)/5(x - 2)/(x + 2)(x - 2) 10x = (x+2)(x-2) We multiply through on the right side to get: 10x = x^2 - 4 Subtract 10x from each side: x^2 - 10x - 4 = 0 This is a quadratic equation. To solve it, [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-10x-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type it in our search engine[/URL] and we get: Speed of the boat in still water =X=5 +- sq. Root of 29 kmph We only want the positive solution: x = 5 + sqrt(29) x = 10.38 [U]Calculate time for upstream journey:[/U] Time for upstream journey = 15/(10.38 - 2) Time for upstream journey = 15/(8.38) Time for upstream journey = [B]1.79[/B] [U]Calculate time for downstream journey:[/U] Time for downstream journey = 15/(10.38 + 2) Time for downstream journey = 15/(12.38) Time for downstream journey = [B]1.21[/B]

A bird was sitting 12 meters from the base of an oak tree and flew 15 meters to reach the top of the tree. How tall is the tree? So we have a [U]right triangle[/U]. Hypotenuse is 15. Base is 12. We want the length of the leg. The formula for a right triangle relation of sides is a^2 + b^2 = c^2 where c is the hypotenuse and a, b are the sides Rearranging this equation to isolate a, we get a^2 = c^2 - b^2 Taking the square root of both sides, we get a = sqrt(c^2 - b^2) a = sqrt(15^2 - 12^2) a = sqrt(225 - 144) a = sqrt(81) a = [B]9 meters[/B]

A rectangular garden is 5 ft longer than it is wide. Its area is 546 ft2. What are its dimensions? [LIST=1] [*]Area of a rectangle is lw. lw = 546ft^2 [*]We know that l = w + 5. [/LIST] Substitute (2) into (1) (w + 5)w = 546 w^2 + 5w = 546 Subtract 546 from each side w^2 + 5w - 546 = 0 Using the positive root in our [URL='http://www.mathcelebrity.com/quadratic.php?num=w%5E2%2B5w-546%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get [B]w = 21[/B]. This means l = 21 + 5. [B]l = 26[/B]

A spherical water tank holds 11,500ft^3 of water. What is the diameter? The tank holding amount is volume. And the volume of a sphere is: V = (4pir^3)/3 We know that radius is 1/2 of diameter: r =d/2 So we rewrite our volume function: V = 4/3(pi(d/2)^3) We're given V = 11,500 so we have: 4/3(pi(d/2)^3) = 11500 Multiply each side by 3/4 4/3(3/4)(pi(d/2)^3) = 11,500*3/4 Simplify: pi(d/2)^3 = 8625 Since pi = 3.1415926359, we divide each side by pi: (d/2)^3 = 8625/3.1415926359 (d/2)^3 = 2745.42 Take the cube root of each side: d/2 = 14.0224 Multiply through by 2: [B]d = 28.005[/B]

Page 43 here, but switch q and r: [URL]http://people.math.gatech.edu/~ecroot/2406_2012/basic_logic.pdf[/URL]

Free Approximate Square Root Using Exponential Identity Calculator - Calculates the square root of a positive integer using the Exponential Identity Method

a^2 + b^2 = c^2 for c Take the square root of each side: c = [B]sqrt(a^2 + b^2)[/B]

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cube root of a number and 7 The phrase [I]a number[/I] means an arbitrary variable, let's call it x: x Cube root of a number means we raise x to the 1/3 power: x^1/3 And 7 means we add 7: [B]x^1/3 + 7[/B]

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difference between 2 positive numbers is 3 and the sum of their squares is 117 Declare variables for each of the two numbers: [LIST] [*]Let the first variable be x [*]Let the second variable be y [/LIST] We're given 2 equations: [LIST=1] [*]x - y = 3 [*]x^2 + y^2 = 117 [/LIST] Rewrite equation (1) in terms of x by adding y to each side: [LIST=1] [*]x = y + 3 [*]x^2 + y^2 = 117 [/LIST] Substitute equation (1) into equation (2) for x: (y + 3)^2 + y^2 = 117 Evaluate and simplify: y^2 + 3y + 3y + 9 + y^2 = 117 Combine like terms: 2y^2 + 6y + 9 = 117 Subtract 117 from each side: 2y^2 + 6y + 9 - 117 = 117 - 117 2y^2 + 6y - 108 = 0 This is a quadratic equation: Solve the quadratic equation 2y2+6y-108 = 0 With the standard form of ax2 + bx + c, we have our a, b, and c values: a = 2, b = 6, c = -108 Solve the quadratic equation 2y^2 + 6y - 108 = 0 The quadratic formula is denoted below: y = -b ± sqrt(b^2 - 4ac)/2a [U]Step 1 - calculate negative b:[/U] -b = -(6) -b = -6 [U]Step 2 - calculate the discriminant ?:[/U] ? = b2 - 4ac: ? = 62 - 4 x 2 x -108 ? = 36 - -864 ? = 900 <--- Discriminant Since ? is greater than zero, we can expect two real and unequal roots. [U]Step 3 - take the square root of the discriminant ?:[/U] ?? = ?(900) ?? = 30 [U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U] Numerator 1 = -b + ?? Numerator 1 = -6 + 30 Numerator 1 = 24 [U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U] Numerator 2 = -b - ?? Numerator 2 = -6 - 30 Numerator 2 = -36 [U]Step 6 - calculate your denominator which is 2a:[/U] Denominator = 2 * a Denominator = 2 * 2 Denominator = 4 [U]Step 7 - you have everything you need to solve. Find solutions:[/U] Solution 1 = Numerator 1/Denominator Solution 1 = 24/4 Solution 1 = 6 Solution 2 = Numerator 2/Denominator Solution 2 = -36/4 Solution 2 = -9 [U]As a solution set, our answers would be:[/U] (Solution 1, Solution 2) = (6, -9) Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution

Free Equation and Inequalities Calculator - Solves an equation or inequality with 1 unknown variable and no exponents as well as certain absolute value equations and inequalities such as |x|=c and |ax| = c where a and c are constants. Solves square root, cube root, and other root equations in the form ax^2=c, ax^2 + b = c. Also solves radical equations in the form asqrt(bx) = c. Also solves open sentences and it will solve one step problems and two step equations. 2 step equations and one step equations and multi step equations

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Explain the relationship between "squaring" a number and finding the "square root" of a number. Use an example to further explain your answer. Squaring a number means raising it to the power of 2 The square root of a number [I]undoes[/I] a square of a number. So square root of x^2 is x x squared is x^2 Let x = 5. x squared = 5^2 = 25 Square root of 25 = square root of 5^2 = 5

Free Factoring and Root Finding Calculator - This calculator factors a binomial including all 26 variables (a-z) using the following factoring principles:
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This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots
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Do you mean x bar? mean of 106 inches and a standard deviation of 10 inches and for sample of size is 25. Determine the mean and the standard deviation of /x If so, x bar equals the population mean. So it's [B]106[/B]. Sample standard deviation = Population standard deviation / square root of n 10/Sqrt(25) 10/5 [B]2[/B]

find the two square roots of 81 When we multiply 9 * 9, we get 81 When we multiply -9 * -9, we get 81 So our two square roots of 81 are: [LIST] [*][B]-9, 9[/B] [/LIST]

Hari planted 324 plants in such a way that there were as many rows of plants as there were number of columns. Find the number of rows and columns. Let r be the number of rows and c be the number of columns. We have the area: rc = 324 Since rows equal columns, we have a square, and we can set r = c. c^2 = 324 Take the square root of each side: [B]c = 18[/B] Which means [B]r = 18[/B] as well. What we have is a garden of 18 x 18.

if i = square root of -1 what is the sum (7 + 3i) + (-8 + 9i) We group like terms, and we get: 7 - 8 + (3 + 9)i Simplifying, we get: [B]-1 + 12i[/B]

If V is the volume of a cube whose side is s, express s in terms of V: We know the Volume (V) of a cube with side length s is: V = s^3 Take the cube root of each side: V^1/3 = (s^3)^1/3 s = [B]V^1/3[/B]

Juan runs out of gas in a city. He walks 30yards west and then 16 yards south looking for a gas station. How far is he from his starting point? Juan is located on a right triangle. We calculate the hypotenuse: 30^2 + 16^2 = Hypotenuse^2 900 + 256 = Hypotenuse^2 Hypotenuse^2 = 1156 Take the square root of each side: [B]Hypotenuse = 34 yards[/B]

K varies inversely with square root of m and directly with the cube of n. [LIST] [*]We take a constant c as our constant of proportionality. [*]The word inversely means we divide [*]The word directly means we multiply [/LIST] [B]k = cn^3/sqrt(m)[/B]

Kamille is calculating the length of diagonal on a picture board and gets a solution of the square root of 58. She needs to buy the ribbon to put across the diagonal of the board, so she estimates that she will need at least 60 inches of ribbon to cover the diagonal. Is she correct? Explain. [URL='https://www.mathcelebrity.com/powersq.php?num=sqrt%2858%29&pl=Calculate']The square root of 58 [/URL]has an answer between 7 and 8. So Kamille is [B]incorrect[/B]. She needs much less than 60 inches of ribbon. She needs less than 8 inches of ribbon.

Free Newton Method Calculator - Calculates the square root of a positive integer using the Newton Method

n^2 + 9 = 34 Subtract 9 from each side: n^2 + 9 - 9 = 34 - 9 n^2 = 25 Take the square root of each side: n = [B]5[/B]

n^2 - 1 = -99/100 Add 1 (100/100) to each side: n^2 - 1 + 1 = -99/100 + 100/100 Cancel the 1's on the left side: n^2 = 1/100 Take the square root of both sides: n = [B]1/10 or -1/10[/B]

n^2 = 1/4 Take the square root of each side: n = [B]1/2[/B]

n^2 = 6&1/4 [URL='https://www.mathcelebrity.com/fraction.php?frac1=6%261%2F4&frac2=3%2F8&pl=Simplify']6&1/4[/URL] = 25/4 n^2 = 25/4 Take the square root of each side: n = [B]5/2 or -5/2[/B]

n^2 = 64 Take the square root of each side: sqrt(n^2) = sqt(64) n = [B]8[/B]

Pleasantburg has a population growth model of P(t)=at^2+bt+P0 where P0 is the initial population. Suppose that the future population of Pleasantburg t years after January 1, 2012, is described by the quadratic model P(t)=0.7t^2+6t+15,000. In what month and year will the population reach 19,200? Set P(t) = 19,200 0.7t^2+6t+15,000 = 19,200 Subtract 19,200 from each side: 0.7t^2+6t+4200 = 0 The Quadratic has irrational roots. So I set up a table below to run through the values. At t = 74, we pass 19,200. Which means we add 74 years to 2012: 2012 + 74 = [B]2086[/B] t 0.7t^2 6t Add 15000 Total 1 0.7 6 15000 15006.7 2 2.8 12 15000 15014.8 3 6.3 18 15000 15024.3 4 11.2 24 15000 15035.2 5 17.5 30 15000 15047.5 6 25.2 36 15000 15061.2 7 34.3 42 15000 15076.3 8 44.8 48 15000 15092.8 9 56.7 54 15000 15110.7 10 70 60 15000 15130 11 84.7 66 15000 15150.7 12 100.8 72 15000 15172.8 13 118.3 78 15000 15196.3 14 137.2 84 15000 15221.2 15 157.5 90 15000 15247.5 16 179.2 96 15000 15275.2 17 202.3 102 15000 15304.3 18 226.8 108 15000 15334.8 19 252.7 114 15000 15366.7 20 280 120 15000 15400 21 308.7 126 15000 15434.7 22 338.8 132 15000 15470.8 23 370.3 138 15000 15508.3 24 403.2 144 15000 15547.2 25 437.5 150 15000 15587.5 26 473.2 156 15000 15629.2 27 510.3 162 15000 15672.3 28 548.8 168 15000 15716.8 29 588.7 174 15000 15762.7 30 630 180 15000 15810 31 672.7 186 15000 15858.7 32 716.8 192 15000 15908.8 33 762.3 198 15000 15960.3 34 809.2 204 15000 16013.2 35 857.5 210 15000 16067.5 36 907.2 216 15000 16123.2 37 958.3 222 15000 16180.3 38 1010.8 228 15000 16238.8 39 1064.7 234 15000 16298.7 40 1120 240 15000 16360 41 1176.7 246 15000 16422.7 42 1234.8 252 15000 16486.8 43 1294.3 258 15000 16552.3 44 1355.2 264 15000 16619.2 45 1417.5 270 15000 16687.5 46 1481.2 276 15000 16757.2 47 1546.3 282 15000 16828.3 48 1612.8 288 15000 16900.8 49 1680.7 294 15000 16974.7 50 1750 300 15000 17050 51 1820.7 306 15000 17126.7 52 1892.8 312 15000 17204.8 53 1966.3 318 15000 17284.3 54 2041.2 324 15000 17365.2 55 2117.5 330 15000 17447.5 56 2195.2 336 15000 17531.2 57 2274.3 342 15000 17616.3 58 2354.8 348 15000 17702.8 59 2436.7 354 15000 17790.7 60 2520 360 15000 17880 61 2604.7 366 15000 17970.7 62 2690.8 372 15000 18062.8 63 2778.3 378 15000 18156.3 64 2867.2 384 15000 18251.2 65 2957.5 390 15000 18347.5 66 3049.2 396 15000 18445.2 67 3142.3 402 15000 18544.3 68 3236.8 408 15000 18644.8 69 3332.7 414 15000 18746.7 70 3430 420 15000 18850 71 3528.7 426 15000 18954.7 72 3628.8 432 15000 19060.8 73 3730.3 438 15000 19168.3 74 3833.2 444 15000 19277.2

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Free Quartic Equations Calculator - Solves quartic equations in the form ax⁴ + bx³ + cx² + dx + e using the following methods:
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r varies directly with s and inversely with the square root of t Varies directly means we multiply Varies inversely means we divide There exists a constant k such that: [B]r = ks/sqrt(t)[/B]

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rs+h^2=1 for h Subtract rs from each side to isolate h: rs - rs + h^2 = 1 - rs Cancel the rs on the left side: h^2 = 1 - rs Take the square root of each side: sqrt(h^2) = sqrt(1 - rs) [B]h = +- sqrt(1 -rs)[/B]

s = tu^2 for u Divide each side by t u^2 = s/t Take the square root of each side [LIST] [*]u = sqrt(s/t) [*]u = -sqrt(s/t) [/LIST] We have two answers due to negative number squared is positive

Six is the principal square root of 36 The two square roots of 36 are: [LIST] [*]+6 [*]-6 [/LIST] The positive square root is known as the principal square root, therefore, this is [B]true[/B].

Solve for h. rs + h^2 = l [U]Subtract rs from each side to isolate h:[/U] rs - rs + h^2 = l - rs [U]Cancel the rs terms on the left side, and we get:[/U] h^2 = l - rs [U]Take the square root of each side:[/U] h = [B]sqrt(l - rs)[/B]

Solve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for v 1/2(2/5) = 1/5 since the 2's cancel r^2/r^2 = 1 So we simplify, and get: mgh=1/2mv^2+1/5(mv^2) for v Divide each side by m, so m's cancel in each term on the left and right side: gh = 1/2v^2 + 1/5(v^2) Combine like terms for v^2 on the right side: 1/2 + 1/5 = 7/10 from our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F2&frac2=1%2F5&pl=Add']fraction calculator[/URL] So we have: gh = 7v^2/10 Multiply each side by 10: 10gh = 7v^2 Now divide each side by 7 10gh/7 = v^2 Take the square root of each side: [B]v = sqrt(10gh/7)[/B]

Square root of 9136 divided by 43 First, [URL='https://www.mathcelebrity.com/powersq.php?num=sqrt%289136%29&pl=Calculate']take the square root of 9136 in our calculator[/URL]: 4 * sqrt(571) Now divide this by 43: [B]4 * sqrt(571) / 43[/B]

square root of the sum of 2 variables The phrase [I]2 variables[/I] means we choose 2 arbitrary variables, let's call them x and y: x, y The sum of 2 variables means we add: x + y Square root of the sum of 2 variables is written as: [B]sqrt(x + y)[/B]

square root of x times the square root of y square root of x: sqrt(x) square root of y: sqrt(y) square root of x times the square root of y [B]sqrt(x) * sqrt(y)[/B]

Free Square Root Table Calculator - Generates a square root table for the first (n) numbers rounded to (r) digits

Free Square Roots and Exponents Calculator - Given a number (n), or a fraction (n/m), and/or an exponent (x), or product of up to 5 radicals, this determines the following:
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The coefficient of determination is found by taking the square root of the coefficient of correlation. True or False [B]FALSE[/B] - It is found by squaring the coefficient of correlation

The IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. a) What is the probability that a randomly person has an IQ between 85 and 115? b) Find the 90th percentile of the IQ distribution c) If a random sample of 100 people is selected, what is the standard deviation of the sample mean? a) [B]68%[/B] from the [URL='http://www.mathcelebrity.com/probnormdist.php?xone=50&mean=100&stdev=15&n=1&pl=Empirical+Rule']empirical rule calculator[/URL] b) P(z) = 0.90. so z = 1.28152 using Excel NORMSINV(0.9)
(X - 100)/10 = 1.21852 X = [B]113[/B] rounded up c) Sample standard deviation is the population standard deviation divided by the square root of the sample size 15/sqrt(100) = 15/10 =[B] 1.5[/B]

Let the number be x. Let the square be x^2. So we have (x)(x^2) = x^3 < 8 Take the cube root of this, we get x = 2

twice a number subtracted from the square root of the same number The phrase [I]a number[/I] means an arbitrary variable, let's call it x: x Twice a number means we multiply x by 2: 2x Square root of the same number: sqrt(x) twice a number subtracted from the square root of the same number [B]sqrt(x) - 2x[/B]

twice the square root of a number increased by 5 is 23 The phrase [I]a number[/I] means an arbitrary variable, let's call it x: x The square root of a number means we raise x to the 1/2 power: sqrt(x) the square root of a number increased by 5 means we add 5 to sqrt(x): sqrt(x) + 5 twice the square root of a number increased by 5 means we multiply sqrt(x) + 5 by 2: 2(sqrt(x) + 5) The phrase [I]is 23[/I] means we set 2(sqrt(x) + 5) equal to 23: [B]2(sqrt(x) + 5) = 23[/B]

Use k as the constant of variation. L varies jointly as u and the square root of v. Since u and v vary jointly, we multiply by the constant of variation k: [B]l = ku * sqrt(v)[/B]

Free Variation Equations Calculator - This calculator solves the following direct variation equations and inverse variation equations below:
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vw^2+y=x for w This is an algebraic expression. Subtract y from each side: vw^2 + y - y = x - y The y's cancel on the left side, so we're left with: vw^2 = x - y Divide each side by v w^2 = (x - y)/v Take the square root of each side: w = [B]Sqrt((x - y)/v)[/B]

X varies directly with the cube root of y when x=1 y=27. Calculate y when x=4 Varies directly means there is a constant k such that: x = ky^(1/3) When x = 1 and y = 27, we have: 27^1/3(k) = 1 3k = 1 To solve for k, we[URL='https://www.mathcelebrity.com/1unk.php?num=3k%3D1&pl=Solve'] type in our equation into our search engine[/URL] and we get: k = 1/3 Now, the problem asks for y when x = 4. We use our variation equation above with k = 1/3 and x = 4: 4 = y^(1/3)/3 Cross multiply: y^(1/3) = 4 * 3 y^(1/3) =12 Cube each side: y^(1/3)^3 = 12^3 y = [B]1728[/B]

You and your friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add three. If your friend's final answer is 53, what was the original number chosen? Let n be our original number. Square the number means we raise n to the power of 2: n^2 Multiply the result by 2: 2n^2 And then add three: 2n^2 + 3 If the friend's final answer is 53, this means we set 2n^2 + 3 equal to 53: 2n^2 + 3 = 53 To solve for n, we subtract 3 from each side, to isolate the n term: 2n^2 + 3 - 3 = 53 - 3 Cancel the 3's on the left side, and we get: 2n^2 = 50 Divide each side of the equation by 2: 2n^2/2 = 50/2 Cancel the 2's, we get: n^2 = 25 Take the square root of 25 n = +-sqrt(25) n = +-5 We are told the number is positive, so we discard the negative square root and get: n = [B]5[/B]