Solving Equations Involving Roots
Hi, and welcome to this video on solving equations involving roots!
Before we dive in, let’s review what a root is. The most common root you will see is a square root. The symbol for a square root is called a radical, and it looks like this: \(\sqrt{ }\). A square root asks the question, what number multiplied by itself (or squared) will give me the number under the radical?
Other roots work in a similar way. When you have a cube root, fourth root, fifth root, etc. it asks the question, what number multiplied by itself three, four, five, times will give me the number under the radical? The way you know how many times to multiply the number by itself is given by a small number put in the hook of the radical symbol. If there is no number there, it is assumed to be a two, or a square root. This is an example of the fourth root of 16. The fourth root of 16, which you can tell is the fourth root by this little 4 in the hook. Is equal to the fourth root of 2 times 2 times 2 times 2. Since there are four two’s we can pull those out because we are looking for the fourth root. And our answer is 2.
Now that we’ve gone over what roots are, let’s learn how to solve equations using them by looking at an example:
Solve the following equation for x.
The square root of x minus 3 equals 4.
The first thing we need to do to get x by itself is to get rid of the square root. We do this by doing the opposite operation to both sides of the equation. The opposite of a square root is a square, so we are going to square both sides.
This gives us x minus 3 is equal to 16.
Now it looks like a regular equation, and we know that all we need to do to get x by itself is add three to both sides.
x equals 19. This gives us our final answer.
That wasn’t too bad! Now let’s look at one with a root other than a two. Let’s try this one. The fifth root of x plus 7 is equal to 2.
In order to get rid of our root, we have to do the opposite operation, which is to put both sides to a power. Since we have a fifth root, we want to put both sides to the fifth power.
This gives us x plus 7 is equal to 32.
Subtracting 7 from both sides gives us our final answer which is x equals 25.
What if the root isn’t the only thing on the left side of our equation? If that’s the case, then we need to follow the order of operations backward to get rid of everything until the root is by itself.
Consider the following example:
Four times the cube root of x minus 9, plus 6, equals 18.
If we go by the backward order of operations, we first have to get rid of all our addition and subtraction, so the first thing we do is subtract 6 from both sides.
Four times the cube root of x minus 9, is equal to 12.
Then we want to divide both sides by 4 to get the root by itself.
The cube root of x minus 9 is equal to 3.
Now we undo the root by cubing both sides.
This gives us x minus 9, is equal to 27.
Finally, we add 9 to both sides to get our answer. x equals 36.
I want to do one more example. This time, try to figure out the answer on your own, then check to see if it matches up with mine.
3 times the square root of x plus 8, minus 12, equals 3.
Think you’ve got it? Let’s check!
First, we want to add 12 to both sides.
3 root x plus 8 is equal to 15.
Then, we divide both sides by 3.
Root x plus 8 is equal to 5.
Now, we are going to square both sides.
This gives us x plus 8 is equal to 25.
And finally, we are going to subtract 8 from both sides to get our final answer.
x is equal to 17.
I hope this review of solving equations involving roots was helpful! Thanks for watching and happy studying!
Solving Equations Involving Roots Practice Questions
Find the value of x in the equation \(\sqrt{x+2}=5\).
To find the answer, solve for x in the equation.
\(\sqrt{x+2}=5\)
To solve for x, perform inverse operations to isolate the variable. Since the opposite of a square root is the exponent 2, raise both sides of the equation to the second power.
\((\sqrt{x+2})^2=5^2\)
The square root of \(x+2\) raised to the second power equals \(x+2\). 5 squared equals 25.
\(x+2=25\)
Next, isolate the variable by performing the inverse operation of adding 2. Since the opposite of adding 2 is subtracting 2, subtract 2 from both sides of the equation.
\(x+2-2=25-2\)
2 minus 2 equals 0, so we are left with x on the left side of the equation. 25 minus 2 equals 23, so x is equal to 23.
\(x=23\)
Check your work by substituting the solution back into the original problem for x. Since 23 plus 2 equals 25, find the square root of 25. The square root of 25 equals 5, so the solution is correct.
\(\sqrt{(23)+2}=5\)
\(\sqrt{25}=5\)
\(5=5\) ✔
Find the value of x in the equation \(\sqrt[4]{x-8}=3\).
To find the answer, solve for x in the equation.
\(\sqrt[4]{x-8}=3\)
To solve for x, perform inverse operations to isolate the variable. Since the opposite of a fourth root is the exponent 4, raise both sides of the equation to the fourth power.
\((\sqrt[4]{x-8})^4=3^4\)
The fourth root of \(x-8\) raised to the fourth power equals \(x-8\). 3 raised to the fourth power equals \(3\times3\times3\times3\), which equals 81.
\(x-8=81\)
Next, isolate the variable by performing the inverse operation of subtracting 8. Since the opposite of subtracting 8 is adding 8, add 8 to both sides of the equation.
\(x-8+8=81+8\)
-8 plus 8 equals 0, so we are left with x on the left side of the equation. 81 plus 8 equals 89, so x is equal to 89.
\(x=89\)
Check your work by substituting the solution back into the original problem for x. Since 89 minus 8 equals 81, find the fourth root of 81. The fourth root of 81 equals 3, so the solution is correct.
\(\sqrt[4]{(89)-8}=3\)
\(\sqrt[4]{81}=3\)
\(\)3=3 ✔
Find the value of x in the equation \(2+\sqrt[3]{8x}=6\).
To find the answer, solve for x in the equation.
\(2+\sqrt[3]{8x}=6\)
To solve for x, perform inverse operations to isolate the variable. Since the opposite of adding 2 is subtracting 2, subtract 2 from both sides of the equation.
\(2+\sqrt[3]{8x}-2=6-2\)
2 minus 2 equals 0, so we are left with \(\sqrt[3]{8x}\) on the left side of the equation. 6 minus 2 equals 4, so write 4 on the right side of the equation.
\(\sqrt[3]{8x}=4\)
Next, do inverse operations again. Since the opposite of a cube root is the exponent 3, raise both sides of the equation to the third power.
\((\sqrt[3]{8x})^3=4^3\)
The cube root of \(8x\) raised to the third power equals \(8x\). 4 raised to the third power equals \(4\times4\times4\), which equals 64.
\(8x=64\)
Next, isolate the variable by performing the inverse operation of multiplying by 8. Since the opposite of multiplying by 8 is dividing by 8, divide both sides of the equation by 8.
\(\frac{8x}{8}=\frac{64}{8}\)
8 divided by 8 equals 1, leaving x by itself on the left side of the equation. 64 divided by 8 equals 8, so x is equal to 8.
\(x=8\)
Check your work by substituting the solution back into the original problem for x. Since 8 times 8 equals 64, find the cube root of 64. The cube root of 64 equals 4, and 2 plus 4 equals 6. Therefore, the solution is correct.
\(2+\sqrt[3]{8(8)}=6\)
\(2+\sqrt[3]{64}=6\)
\(2+4=6\)
\(6=6\) ✔
Find the value of x in the equation \(\frac{(\sqrt[3]{x+100})-2}{3}=1\)
To find the answer, solve for x in the equation.
\(\frac{(\sqrt[3]{x+100})-2}{3}=1\)
To solve for x, perform inverse operations to isolate the variable. Since the opposite of dividing by 3 is multiplying by 3, multiply both sides of the equation by 3.
\(\frac{(\sqrt[3]{x+100})-2}{3}\times3=1\times3\)
Dividing by 3 and multiplying by 3 cancel each other out, so we are left with \((\sqrt[3]{x+100})-2\) on the left side of the equation. Since 1 times 3 equals 3, write 3 on the right side of the equation.
\((\sqrt[3]{x+100})-2=3\)
Next, do the inverse operation of subtracting 2. Since the opposite of subtracting 2 is adding 2, add 2 to both sides of the equation.
\((\sqrt[3]{x+100})-2+2=3+2\)
-2 plus 2 equals 0, so we are left with \(\sqrt[3]{x+100}\) on the left side of the equation. 3 plus 2 equals 5, so write 5 on the right side of the equation.
\(\sqrt[3]{x+100}=5\)
Next, do inverse operations again. Since the opposite of a cube root is the exponent 3, raise both sides of the equation to the third power.
\((\sqrt[3]{x+100})^3=5^3\)
The cube root of \(x+100\) raised to the third power equals \(x+100\). 5 raised to the third power equals \(5\times5\times5\), which equals 125.
\(x+100=125\)
Next, isolate the variable by performing the inverse operation of adding 100. The opposite of adding 100 is subtracting 100, so subtract 100 from both sides of the equation.
\(x+100-100=125-100\)
100 minus 100 equals 0, so we are left with x on the left side of the equation. 125 minus 100 equals 25, so x is equal to 25.
\(x=25\)
Check your work by substituting the solution back into the original problem for x. Since 25 plus 100 equals 125, find the cube root of 125, which is 5. 5 minus 2 equals 3, and 3 divided by 3 equals 1. Therefore, the solution is correct.
\(\frac{(\sqrt[3]{x+100})-2}{3}=1\)
\(\frac{(\sqrt[3]{125})-2}{3}=1\)
\(\frac{(5)-2}{3}=1\)
\(\frac{3}{3}=1\)
\(1=1\) ✔
Find the value of x in the equation \(6+2\times \sqrt[4]{16x}-8=2\).
To find the answer, solve for x in the equation.
\(6+2\times \sqrt[4]{16x}-8=2\)
To solve for x, perform inverse operations to isolate the variable. Since the opposite of subtracting 8 is adding 8, add 8 to both sides of the equation.
\(6+2\times \sqrt[4]{16x}-8+8=2+8\)
-8 plus 8 equals 0, so the eights on the left side of the equation cancel out. 2 plus 8 equals 10, so write 10 on the right side of the equation.
\(6+2\times \sqrt[4]{16x}=10\)
Next, do the inverse operation of adding 6. Since the opposite of adding 6 is subtracting 6, subtract 6 from both sides of the equation.
\(6-6+2\times \sqrt[4]{16x}=10-6\)
6 minus 6 equals 0, so the sixes on the left side of the equation cancel out. 10 minus 6 equals 4, so write 4 on the right side of the equation.
\(2\times \sqrt[4]{16x}=4\)
Next, do the inverse operation of multiplying by 2. Since the opposite of multiplying by 2 is dividing by 2, divide both sides of the equation by 2.
\(\frac{2\times \sqrt[4]{16x}}{2}=\frac{4}{2}\)
2 divided by 2 equals 1, so we are left with 416x on the left side of the equation. 4 divided by 2 equals 2, so write 2 on the right side of the equation.
\(\sqrt[4]{16x}=2\)
Next, do inverse operations again. Since the opposite of a fourth root is the exponent 4, raise both sides of the equation to the fourth power.
\((\sqrt[4]{16x})^4=2^4\)
The fourth root of \(16x\) raised to the fourth power is \(16x\). 2 raised to the fourth power is \(2\times2\times2\times2\), which equals 16.
\(16x=16\)
Next, isolate the variable by doing the inverse operation of multiplying by 16. Since the opposite of multiplying by 16 is dividing by 16, divide both sides of the equation by 16.
\(\frac{16x}{16}=\frac{16}{16}\)
16 divided by 16 equals 1, so we are left with x on the left side of the equation. 16 divided by 16 equals 1, so x is equal to 1.
\(x=1\)
Check your work by substituting the solution back into the original problem for x. Since 16 times 1 equals 16, find the fourth root of 16, which is 2. Next, solve \(6+2\times2-8\). According to the Order of Operations, multiply before adding or subtracting. Since 2 times 2 equals 4, rewrite the problem using this product. 6 plus 4 equals 10, and 10 minus 8 equals 2. Therefore, the solution is correct.
\(6+2\times \sqrt[4]{16(1)}-8=2\)
\(6+2\times \sqrt[4]{16}-8=2\)
\(6+2\times2-8=2\)
\(6+4-8=2\)
\(10-8=2\)
\(2=2\) ✔