The denominator will turn into a difference of squares. Eliminate the parentheses with the squared first. Then combine variables and add or subtract exponents. Then get rid of parentheses first, by pushing the exponents through. Remember that the bottom of the fraction is what goes in the root, and we typically take the root first. We could also put this in our calculator! Flip fraction first to get rid of negative exponent. Flip the fraction, and then do the math with each term separately.
Simplify the roots both numbers and variables by taking out squares. Then, combine like terms, where you need to have the same root and variables. Then we applied the exponents, and then just multiplied across. Put it all together, combining the radical. Move all the constants numbers to the right. First, divide both sides by 2 ; always try to simplify before solving with radicals.
We can raise both sides to the same number. We want to raise both sides to the reciprocal of the exponent on the left , so it turns into a 1. Remember that when you raise an exponent to another exponent, you can multiply the two exponents. This is a neat trick! Since we have the cube root on each side, we can simply cube each side. Since we can never square any real number and end up with a negative number, there is no real solution for this equation.
Since we have square roots on both sides, we can simply square both sides to get rid of them. We have to make sure we square the 4 too. In the case of a fractional exponent on the left near the variable , if the even number is on the top of the fraction, you have to take the positive and negative solutions.
This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. The exponent is distributed in the same way. Look at that—you can think of any number underneath a radical as the product of separate factors , each underneath its own radical.
This rule is important because it helps you think of one radical as the product of multiple radicals. Using the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical.
Rearrange factors so the integer appears before the radical, and then multiply. This is done so that it is clear that only the 7 is under the radical, not the 3.
The following video shows more examples of how to simplify square roots that do not have perfect square radicands. Before we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand. This looks like it should be equal to x , right? Where are they equal? Where are they not equal? We will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.
Take the square root of each radical. One tip for knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. If the exponent is odd — including 1 — add an absolute value.
This applies to simplifying any root with an even index, as we will see in later examples. In the following video you will see more examples of how to simplify radical expressions with variables. We will show another example where the simplified expression contains variables with both odd and even powers.
Because x has an odd power, we will add the absolute value for our final solution. In our next example we will start with an expression written with a rational exponent. You will see that you can use a similar process — factoring and sorting terms into squares — to simplify this expression. We can use the same techniques we have used for simplifying square roots to simplify higher order roots. You can use fractional exponents that have numerators other than 1 to express roots, as shown below.
Notice any patterns within this table? To rewrite a radical using a fractional exponent, the power to which the radicand is raised becomes the numerator and the root becomes the denominator. Writing Fractional Exponents. Any radical in the form can be written using a fractional exponent in the form.
The relationship between and works for rational exponents that have a numerator of 1 as well. For example, the radical can also be written as , since any number remains the same value if it is raised to the first power. You can now see where the numerator of 1 comes from in the equivalent form of.
One method of simplifying this expression is to factor and pull out groups of a 3 , as shown below in this example. Rewrite by factoring out cubes. Write each factor under its own radical and simplify. You can also simplify this expression by thinking about the radical as an expression with a rational exponent, and using the principle that any radical in the form can be written using a fractional exponent in the form.
Note that rational exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.
Both simplification methods gave the same result, a 2. Rewrite the radical using rational exponents. Use the rules of exponents to simplify the expression. Change the expression with the rational exponent back to radical form. Again, the alternative method is to work on simplifying under the radical by using factoring. For the example you just solved, it looks like this. Which of the expressions below is equal to the expression when written using a rational exponent? The problem asks for an expression with rational, or fractional, exponents.
This answer is for simplifying the radical expression. The correct answer is. I raise something to an exponent and then raise that whole thing to another exponent, I can just multiply the exponents. Five over six. Let's do one more of these. The following equation is true for x greater than zero, and d is a constant. Alright, this is interesting. And I forgot to tell you in the last one, but pause this video as well and see if you can work it out on Well, here, let's just start rewriting the root as an exponent.
So, I can rewrite the whole thing. And if I have one over something to a power, that's the same thing as that something raised to the negative of that power. And so, that is going to be equal to x to the d.
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