Exponents and Radicals in Algebra

This section covers:

We briefly talked about exponents in the Powers, Exponents, Radicals (Roots) and Scientific Notation section, but we need to go a little bit further in depth and talk about how to do algebra with them.  Note that we’ll see more radicals in the Solving Radical Equations and Inequalities section, and we’ll talk about Factoring with Exponents, and Exponential Functions in the Exponential Functions section.

Remember that exponents, or “raising” a number to a power, are just the number of times that the number (called the base) is multiplied by itself.

Radicals (which comes from the word “root” and means the same thing) means undoing the exponents, or finding out what numbers multiplied by themselves comes up with the number.

So we remember that  \(\sqrt{25}=5\), since  \(5\times 5=25\). Note that we have to remember that when taking the square root (or any even root), we always take the positive value (just memorize this).

Introducing Exponents and Radicals (Roots) with Variables

But now that we’ve learned some algebra, we can do exponential problems with variables in them!  So we have  \(\sqrt{{{x}^{2}}}=x\)  (actually  \(\sqrt{{{x}^{2}}}=\left| x \right|\)  since x can be negative) since  \(x\,\times \,x={{x}^{2}}\).   We also learned that taking the square root of a number is the same as raising it to  \(\frac{1}{2}\), so  \({{x}^{\frac{1}{2}}}=\sqrt{x}\).   Also, remember that when we take the square root, there’s an invisible 2 in the radical, like this:  \(\sqrt[2]{x}\).

Also note that what’s under the radical sign is called the radicand (x in the previous example), and for the nth root, the index is n (2, in the previous example, since it’s a square root).

With a negative exponent, there’s nothing to do with negative numbers!  You move the base from the numerator to the denominator (or denominator to numerator) and make it positive!  So if you have a base with a negative number that’s not a fraction, put 1 over it and make the exponent positive.

And if the negative exponent is on the outside parentheses of a fraction, take the reciprocal of the fraction and make the exponent positive.  Some examples:   \(\displaystyle {{x}^{-2}}={{\left( \frac{1}{x} \right)}^{2}}\)  and   \(\displaystyle {{\left( \frac{y}{x} \right)}^{-4}}={{\left( \frac{x}{y} \right)}^{4}}\).

Just a note that we’re only dealing with real numbers at this point; later we’ll learn about imaginary numbers, where we can (sort of) take the square root of a negative number.

Properties of Exponents and Radicals

So remember these basic rules:

Basic Exponential and Radical Rules

In algebra, we’ll need to know these and many other basic rules on how to handle exponents and roots when we work with them.  Here are the rules/properties with explanations and examples.  In the “proof” column, you’ll notice that we’re using many of the algebraic properties that we learned in the Types of Numbers and Algebraic Properties section, such as the Associate and Commutative properties.

Unless otherwise indicated, assume numbers under radicals with even roots are positive, and numbers in denominators are nonzero.

Exponent Rules Radical Rules Rationalizing Fractions Denominators

I know this seems like a lot to know, but after a lot of practice, they become second nature.  You will have to learn the basic properties, but after that, the rest of it will fall in place!

Putting Exponents and Radicals in the Calculator

We can put exponents and radicals in the graphing calculator, using the carrot sign  (^) to raise a number to something else, the square root button to take the square root, or the MATH button to get the cube root or nth root.   Be careful though, because if there’s not a perfect square root, the calculator will give you a long decimal number that’s not the “exact value”.  The “exact value” would be the answer with the root sign in it!

Here are some exponent and radical calculator examples:

Exponents and Radicals in the Graphing Calculator Rationalizing Radicals

Before we work example, let’s talk about rationalizing radical fractions.  In math, sometimes we have to worry about “proper grammar”.  When radicals, it’s improper grammar to have a root on the bottom in a fraction – in the denominator.  To fix this, we multiply by a fraction with the bottom radical(s) on both the top and bottom (so the fraction equals 1); this way the bottom radical disappears.  Neat trick!

Here are some examples:

Rationalizing Denominators Radicals Roots

Simplifying Exponential Expressions

There are five main things you’ll have to do to simplify exponents and radicals.  For the purpose of the examples below, we are assuming that variables in radicals are non-negative, and denominators are nonzero.

  • get rid of parentheses ().  Remember that when an exponential expression is raised to another exponent, you multiply exponents.  Also remember when you are multiplying numbers and variables and the whole thing is raised to an exponent, you can remove parentheses and “push through” the exponent.  Example:  
  • combine bases to combine exponents.  You should add exponents of common bases if you multiplying, and subtract exponents of common bases if you are dividing (you can subtract “up”, or subtract “down” to get the positive exponent as you’ll see).  Sometimes you have to match the bases first in order to combine exponents – see last example below.  Examples:  
  • get rid of negative exponents. To get rid of negative exponents, you can simply move a negative exponent in the denominator to the numerator and make it positive, or vice versa.  Examples:  
  • simplify any numbers (like = 2).   Also, remember to simplify radicals by taking out any factors of perfect squares (under a square root), cubes (under a cube root), and so on.  Example:  
  • combine any like terms.   If you’re adding or subtracting terms with the same numbers (coefficients) and/or variables, you can put these together.  Almost think of a radical expression (like ) like another variable.   Example:    .  Remember that, for the variables, we can divide the exponents inside by the root index – if it goes in exactly, we can take the variable to the outside; if there are any remainders, we have to leave the variables under the root sign.  For example, ,  since 5 divided by 3 is 1, with 2 left over (for the x), and 12 divided by 3 is 4 (for the y).

Now let’s put it altogether.  Here are some (difficult) examples.  Just remember that you have to be really, really careful doing these!

Simplify Exponents Negative Exponents Fractional Exponents

Here are even more examples.  Assume variables under radicals are non-negative.

More Examples of Simplifying Radical Expressions

If we don’t assume variables under the radicals are non-negative, we have to be careful with the signs and include absolute values for even radicals.  Here’s an example:

Solving Exponent Equations with Even Radical and Negative

For all these examples, see how we’re doing the same steps over and over again – just with different problems?  If you don’t get them at first, don’t worry; just try to go over them again.  You’ll get it!  And don’t forget that there are many ways to arrive at the same answers!

Solving Exponential and Radical Equations

(We’ll see more of these types of problems here in the Solving Radical Equations and Inequalities section).

Now that we know about exponents and roots with variables, we can solve equations that involve them.

The trick is to get rid of the exponents, we need to take radicals of both sides, and to get rid of radicals, we need to raise both sides of the equation to that power.

You have to be a little careful, especially with even exponents and roots (the “evil evens”), and also when the even exponents are on the top of a fractional exponent (this will become the root part when we solve).

When we solve for variables with even exponents, we most likely will get multiple solutions, since when we square positive or negative numbers, we get positive numbers.  Also, all the answers we get may not work, since we can’t take the even roots of negative numbers.

So it’s a good idea to always check our answers when we solve for roots (especially even roots)!

Let’s first try some equations with odd exponents and roots, since these are a little more straightforward.  (Notice when we have fractional exponents, the radical is still odd when the numerator is odd).

Solving Exponential Equations with Odd Exponents

Now let’s solve equations with even roots.  Note that when we take the even root (like the square root) of both sides, we have to include the positive and the negative solutions of the roots.  (Notice when we have fractional exponents, the radical is still even when the numerator is even.)

Again, when the original problem contains an even root sign, we need to check our answers to make sure we have end up with no negative numbers under the even root sign (no negative radicands).

Also, if we have squared both sides (or raised both sides to an even exponent), we need to check our answers to see if they work.

The solutions that don’t work when you put them back in the original equation are called extraneous solutions.

Again, we’ll see more of these types of problems in the Solving Radical Equations and Inequalities section.

Solving Exponential Equations with Even Exponents

And here’s one more where we’re solving for one variable in terms of the other variables:

Solving Exponential Equations Even Roots Variables Only

Note:  You can also check your answers using a graphing calculator by putting in what’s on the left of the = sign in “” and what’s to the right of the equal sign in “”.    You can then use the intersection feature to find the solution(s); the solution(s) will be what x is at that point.

Here are those instructions again, using an example from above:
Solving Exponential Equations with Graphing Calculator

Solving Simple Radical Inequalities

Note again that we’ll see more problems like these, including how to use sign charts with solving radical inequalities here in the Solving Radical Equations and Inequalities section.

Just like we had to solve linear inequalities, we also have to learn how to solve inequalities that involve exponents and radicals (roots).  We’ll do this pretty much the same way, but again, we need to be careful with multiplying and dividing by anything negative, where we have to change the direction of the inequality sign.

We also have to be careful that our answer still keeps what’s under an even radical to be positive, so we have to create another inequality and set what’s under the even radical to greater than or equal to 0, solve for x, and take the intersection of both solutions.  The reason we take the intersection of the two solutions is because both must work.

With odd roots, we don’t have to worry – we just raise each side that power, and solve!

Here are some examples:

Solving Radical Equations with Inequalities

Learn these rules, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Introduction to Multiplying Polynomials – you are ready!

48 thoughts on “Exponents and Radicals in Algebra

  1. I came looking for help solving for x in an equation where all the instances of x aren’t roots (and the one that is is in denominator):
    1+1/2 x^(-1/2)-2x

    • I’m not sure exactly how to parse your expression, but you’ll probably have to rationalize the fraction with x^1/2 and then use common denominators to add the three expressions. I haven’t gotten into rationals yet (variables in the denominator) – it will be later. But my best guess for your problem would be 1 + 1/(2sqrtx) – 2x = 1 + sqrt(x)/(2x) – 2x = (2x + sqrt(x) – 4x^2)/2x. Does that make sense?

    • The 8th root of y squared would be (y^2)^(1/8) or y^(1/4) or y raised to the 1/4, or the 4th root of y. Remember that when you raise something to something and raise it again, you multiply the exponents. For example (16^(1/4))^2 = 16^(1/4 x 2) = 16^(1/2) = square root of 16 = 4. Does that make sense?

  2. i am having a horrible amount of trouble with dividing and simplifying radicals. Ex: (cuberoot 10y^8)/(cuberoot 27x^9) This is an entire section in my algebra course. please help!

    • Thanks for writing! With these types of problems, just see if you can take out any cube roots, and you can! So we’ll be left with y^2cuberoot(10y^2)/(3x^3). Do you see how with numbers, we just take out any cube roots, and with letters (variables), we divide 3 into the exponents, and leave remainders inside the cube roots? Does that make sense? Lisa

    • Thanks for writing! The square root of 12y^5 would be sqrt(12y^5) = sqrt(12) x y^(5/2) = 2sqrt(3) x y ^ (5/2). Does that make sense? When we take roots, we put the root on the bottom of the fraction of the exponent. Lisa

    • Here’s how I would solve: 4 = 25t^4; 4/25 = t^4; t^2= +/- 2/5; t = +/- sqrt(2/5), or +/- sqrt(10)/5, if you rationalize. Does that make sense? Lisa

      • Okay that’s the answer I keep getting. I think I’m missing something from the expression. Thank you so much for your quick response!

    • Thanks for writing! Sqrt(x^2 – y^2) = sqrt(x^2(1 – y^2/x^2)) (multiply it back through to see why!), which is sqrt(x^2)*sqrt(1 – y^2/x^2) = x*(1 – y^2/x^2). Does that make sense? Lisa

      • I think I get it now, thanks.
        So basically, taking the x^2 factor out, I’m dividing both the x^2 and y^2 factors in the brackets by x^2, giving me 1-y^2/x^2 inside the radical, and I need to multiply this (I mean sqrt(1-y^2/x^2)) by sqrt(x^2) (which is x), obviously in order to obtain the same result.

        Now you’ve gone through it, it makes complete sense, and I’m kicking myself that I didn’t understand it. Thank you very much for the answer 😉

  3. I have another question for you, and I hope that I can ask it in a way that makes sense 😉

    If a^2 = b^2 + c^2, then a does not equal b + c.
    I know this to be true, because I can demonstrate it geometrically with triangles, and numerically just by putting numbers in and seeing what happens – kind of trial and error.
    For instance 5^2 = 3^ + 4^2, but 5 doesn’t equal 3 + 4.

    What I’m looking for, though, is the algebraic proof or law that explains why, if a^2 = b^2 + c^2, then it is impossible for it to also be true that a = b + c.
    In essence that the square root of the sum of 2 squares cannot equal the sum of the individual square roots of those 2 squares.

    It’s something I know to be true, but I just don’t know how that would look written as a proof – as some algebraic equation which can be shown to be true in all circumstances.

    If this question is nonsensical, though, could you tell me why?
    Though I have a feeling you’ll again explain it, it will again be incredibly simple, and I’ll again kick myself for trying to do maths after half a bottle of wine.

    Thank you again for your time 😉

    • Thanks for writing again. Here’s how I’d “prove” it: Let a + b = c (I’m trying to prove it a^2 + b^2 doesn’t equal c^2). Then a = c – b. So substitute to show that a^2 + b^2 doesn’t equal c^2: (c – b)^2 + b^2 = c^2 – 2cb + b^2 +b^2. If this equal c^2, then c^2 – 2cb + 2b^2 = c^2, which would mean that 2b^2 = 2cb, which would mean that b would have to equal c. This would work only if a = 0.

      Does this make sense? Lisa

      • Thank you so much.
        I had to write it out vertically to get the gist. Though I think you meant c^2 – 2cb + 2b^2 = c^2, instead of c^2 – 2cb + b^2 = c^2, am I correct? It seemed to agree with your statement that it would then require 2bc to equal 2b^2, and this for b to equal c , and thus to make sense that this is only possible if a is 0 – thus proving that it is both only possible if a = 0 and if impossible if all three terms represent different integer values.

        Thank you again – your impeccable guidance has really helped 😉

        • I’d like to provide another algebraic proof, similar to Lisa’s proof.

          Problem: If a^2 = b^2 +c^2 Prove: a is not = b+c

          Solution: Let a = b+c
          If we substitute this into the original equation and find it false,
          then a cannot = b+c.

          Given: a^2 = b^2 + c^2
          (b+c)^2 = b^2 + c^2
          b^2 + 2bc + c^2 = b^2 + c^2 Only true if b or c = 0

  4. Can someone please solve these two equations?

    Simplify the expression. Do not assume the variables represent positive numbers.

    sqrt(45 a^3 b^2)=

    Rationalize the denominator in the following expression.

    • Thanks for writing!! sqrt(45a^3b^2) = 3abs(a)abs(b)sqrt(5a). Does that make sense? If variables can be negative, when we take an even root of them, we take the positive number out; thus, the absolute values.
      For the second one, I’m not sure if the 7 is under the root, but I did the problem like it’s not. So I multiplied by sqrt(3)/sqrt(3) to get 2sqrt(3)/21. Does that make sense? Lisa

  5. I got the same answer for the first problem, but I interpreted the second problem differently so I got a different answer. I took root(3) to mean the 3rd root. Usually, sqrt means the square root or root(2). So, for the second problem, I assumed we started with: 2 over the third root of 7.
    I multiplied the numerator and denominator by 7^(2/3) to get my answer:
    I will create a video, called “Radical Example 02” on my website. Check it out!
    Go to: http://www.paulkUSA.com Then, click on Math Videos

  6. Hello Lisa
    Reference to your problem (fourthroot of 64a^7b^8).”Assume variables under radical are non negative”
    Your reducing the fmourth root of a^7 to ( a (fourth root of a^3) got me to thinking about finding the lelnth
    root of any x^m root. I figured out a rather easy method of finding the any nth root of any a^m. While I thought this was sheer brilliance on my part, I confess I haven’t a clue how to explain it.
    Example; the third root of a^7 = reduces to (a (third root of a^3) a=2, ans: 5.03968
    Reducing this further, (a^2 (times third root of a) a=2, ans: 5.03968
    Reducing this further (a^3 (times third root of a^(-2)) a=2 ans: 5.03968
    Reducing this further (a^4 (times third root of a^(a^(-5)) a=2 ans 5.039698
    I can find the nth root of any a^m root. I simply add an exponent to the a, a^2, a^3 etc. multiply it times the a^m and subtract the original exponent. That is, a^4 (3 sqrt = 4 x 3 -7 = sqrt a^(-5).
    Could you explain why this is true?
    Again I am humbled by my lack of real understanding. I blame these tremendous math programs that solve the most esoteric problems but do not explain, beyond, rrmaybe a few elementary steps. I wonder if you have ever considered compiling a book of your work? It would be an extraordinary contribution.
    Thank you Lisa.

    • Manuel,
      Thanks so much for writing! You found an amazing result – I’m so so impressed. I think what you’re doing actually is subtracting integers (1, 2, 3, etc) from the exponent of a. The nth root of a^m is actually a^(m/n). We can write m/n into 2 fractions, for example, n/n + (m-n)/n, which would produce your first result, where n = 3 and m = 7 (a^1 * a^1/3), and so on. Does that make sense? Try the other numbers and see if it does.

      Do you really think I should put all this in a book? I’m thinking about creating a series of e-books – maybe it’s a good idea!

  7. Hi,

    Did you forget to include the negative possibility for the following problem: (y+2)^(2/3) = 4
    y = 6 and Y = -10


    • Thank you SO MUCH for finding this error; you are absolutely correct! I have switched the problem to make it with an odd exponent. Thanks again for could you proof read the entire site? lol Lisa

  8. Can u help me solve this problems..with step plss

    1.)8th root of 16x^8y^4
    2.)6th root of 1/27x^3y^9
    3.)1/4th root of 72xy^8
    4.)8/5 root of 2 + 2 root of 5

    • Thanks for writing: Here’s what I’d get: 1) (2^4)^(1/8)*x*(y^4)^(1/8) = sqrt(2)xy^1/2 2) 3^(-1/2)x^(1/2)y^(3/2) 3) 2^(12)*3^(8)x^4y^32 4) 3.213 (had to put in calculator 😉 These are tough – the idea is you want to raise the expression by the reciprocal of the root. Does that make sense? Lisa

    • I’m not sure if this what you’re asking, but it does turn out that the cube root of 2 can be expressed as 16^(1/12) and the 4th root of 3 can be expressed as 27^(1/12). Is that what you mean? Lisa

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