Equations with Powers, Roots, and Radicals
Read this topicExplanations 4
- (Video) Solving an Equation Containing Two Radicals by PatrickJMT Here's a video by PatrickJMT where he goes through a problem with radicals on both sides of the equation sign. …
By Alex Federspiel, Amy Feraco, Anusha Rahman
- (Video) Radical Equations by yaymath Here's a video by yaymath showing you how to solve a radical equation where the radical is on both sides. Summary\large{\text{Summary}}Summa…
By Alex Federspiel, Amy Feraco, Anusha Rahman
- Example Walkthrough Note that there is a difference between: Eliminating radicals in the denominator Eliminating radicals in an equation Remember, we can manipulate an expressio…
By Christine, Joshua Siktar, Alex Federspiel, Anusha Rahman
- Before diving into this lesson, it may be helpful to review how to solve exponential equations and exponent properties. That mindset may be helpful when tackling this next lesso…
By Christine, Joshua Siktar, Caroline K, Alex Federspiel, Anusha Rahman
Problems 36
- Solve for xxx. x+1=2x−3\sqrt{x+1}=\sqrt{2x-3}x+1=2x−3
- Solve for xxx. x+9=x+1\sqrt{x+9}=\sqrt{x}+1x+9=x+1
- Solve for xxx. 2x+11=8x+2−3\sqrt{2x+11}=\sqrt{8x+2}-32x+11=8x+2−3
- Solve for xxx. x2+8=22x−1\sqrt{x^2+8}=2\sqrt{2x-1}x2+8=22x−1
- Solve for xxx. x−4=9x\sqrt{x}-4=\sqrt{9x}x−4=9x
- Solve for xxx. x−6=7x−22\sqrt{x-6}=\sqrt{7x}-22x−6=7x−22
- Solve 5x+6+3=3x+3+4\sqrt{5x+6}+3=\sqrt{3x+3}+45x+6+3=3x+3+4.
- Find the smallest integer nnn such that n−n−99<1.\sqrt{n} - \sqrt{n - 99} < 1.n−n−99<1. (Hint: First solve the equation x−x−99=1\sqrt{x} - \sqrt{x - 99} = 1x−x−99=1.)
- Solve the equation x−4=x−2\sqrt{x-4} = \sqrt{x} - 2x−4=x−2 for xxx.
- The equation x2=1x^2 = 1x2=1 has two real solutions, −1-1−1 and 111. How many real solutions does the equation x3=1x^3 = 1x3=1 have?
- List all solutions to 3−x=x3−x\sqrt{3-x} = x \sqrt{3-x}3−x=x3−x.
- Solve for xxx in the equation x−1−x+1+1=0\sqrt{x-1} - \sqrt{x+1} + 1 = 0x−1−x+1+1=0. Express your answer as a common fraction.
- Let f(x)=xf(x) = \sqrt{x}f(x)=x. If ccc is a real number satisfying f(c)=c4f(c) = \frac{c}{4}f(c)=4c , then what is f(2c+4)f(2c+4)f(2c+4)?
- It turns out 2+3=a+bc\sqrt{2 + \sqrt{3}} = \frac{a + \sqrt{b}}{\sqrt{c}}2+3=ca+b, where a,b,ca,b,ca,b,c are positive integers with no common factor, and b,cb,cb,c are "squa…
- How many nonzero real numbers satisfy xx=xx^x = xxx=x?
- Given the function k(t)=t3+2k(t)=t^3+2k(t)=t3+2: a) Evaluate k(2)k(2)k(2). b) Solve k(t)=1k(t) = 1k(t)=1.
- Solve for xxx. x−3=12\\\sqrt{x-3}=12x−3=12
- Solve for xxx. x+5=6x\\\sqrt{x+5}=6\sqrt{x}x+5=6x
- Solve for xxx: x−2=2\sqrt{x-2}=\sqrt{2}x−2=2
- Solve for xxx: x+4=3+1\sqrt{x+4}=\sqrt{3}+1x+4=3+1
- Solve for xxx: x3−1=3+1\frac{\sqrt{x}}{\sqrt{3}-1 }=\sqrt{3}+13−1x=3+1
- What is the multiplicity of the root x=2x=\sqrt{2}x=2 in the function y=x−2\sqrt{y}=x-\sqrt{2}y=x−2
- What is the multiplicity of the root x=−3x=-\sqrt{3}x=−3 in the function y=(x+3)2\sqrt{y}=(x+\sqrt{3})^2y=(x+3)2
- Can you fully rationalize an equation of the form x=a+b\sqrt{x}=\sqrt{a}+bx=a+b by squaring?
- Can you fully rationalize an equation of the form x+b=a\sqrt{x+b}=\sqrt{a}x+b=a by squaring? Assume a,b,a, b,a,b, and xxx are all rational numbers.
- Can you fully rationalize an equation of the form x=ab\sqrt{x}=\frac{a}{\sqrt{b}}x=ba by squaring?
- Solve for yyy: cos(x)+y=ln(z)\cos(x)+\sqrt{y}=\ln(z)cos(x)+y=ln(z)
- Here's an equation for error propagation in circuits: αR=R(αVV)2+(αII)2.\alpha_R = R\sqrt{\left(\frac{\alpha_V}{V}\right)^2 + \left(\frac{\alpha_I}{I}\right)^2}.αR=R(VαV)2+(I…
- Solve for xxx: −729x63=−225\sqrt[3]{-729x^6} = -2253−729x6=−225
- Solve for xxx: 325×8x33=52\sqrt[5]{32} \times \sqrt[3]{8x^3} = 52532×38x3=52
- Find the positive solution to x2−3=3−x.\sqrt{x^2 - 3} = \sqrt{3 - x}.x2−3=3−x.
- Solve the equation for x:x:x: 3x+2+4x+8=10.3\sqrt{x + 2} + \sqrt{4x + 8} = 10.3x+2+4x+8=10.
- Solve the equation for xxx when x≥−2:x \geq -2:x≥−2: 7x−1+2x2+4x+4=2x+11.7\sqrt{x - 1} + 2\sqrt{x^2 + 4x + 4} = 2x + 11.7x−1+2x2+4x+4=2x+11.
- Which of the following is a nonnegative solution to 36x+5=24x+20+15x+5?3\sqrt{6x + 5} = \sqrt{24x + 20} + 15x + \sqrt{5}?36x+5=24x+20+15x+5?
- Solve the following equation for xxx, where x≠0x \neq 0x=0. 4x2+8x=2x+316x\sqrt{4x^2+8x} = 2\sqrt{x} + 3 \sqrt{16x}4x2+8x=2x+316x
- Solve for xxx. x+7=1−x\sqrt{x+7}=1-\sqrt xx+7=1−x