Learning Objectives

(9.2.1) – Define and identify a radical expression(9.2.2) – convert radicals to expressions through rational exponents(9.2.3) – convert expressions through rational exponents to your radical equivalent(9.2.4) – reasonable exponents whose molecule is no equal to one(9.2.5) – leveling Radical ExpressionsSimplify radical expressions using factoringSimplify radical expressions using reasonable exponents and the regulations of exponents
(9.2.1) – Define and also identify a radical expression

Square roots are most regularly written making use of a radical sign, like this, \sqrt4. But there is another means to represent them. You deserve to use reasonable exponents instead of a radical. A rational exponent is one exponent that is a fraction. Because that example, \sqrt4 can be written as 4^\tfrac12.

You are watching: 25 3/2 in radical form

Can’t imagine increasing a number to a rational exponent? They might be hard to obtain used to, but rational exponents have the right to actually assist simplify part problems. Creating radicals with rational exponents will certainly come in handy once we comment on techniques for simplifying more complex radical expressions.

Radical expressions room expressions that contain radicals. Radical expressions come in plenty of forms, from simple and familiar, such as \sqrt16, to rather complicated, as in \sqrt<3>250x^4y

(9.2.2) – transform radicals to expressions with rational exponents

Radicals and fractional exponents are alternating ways of expressing the same thing. In the table below we display equivalent ways to express radicals: with a root, v a rational exponent, and also as a primary root.

Radical Form

Exponent Form

Principal Root

\sqrt16 16^\tfrac124
\sqrt25 25^\tfrac125
\sqrt100 100^\tfrac1210

Let’s look at some more examples, but this time with cube roots. Remember, cubing a number raises it come the strength of three. Notification that in the examples in the table below, the denominator the the reasonable exponent is the number 3.

Radical Form

Exponent Form

Principal Root

\sqrt<3>8 8^\tfrac132
\sqrt<3>8 125^\tfrac135
\sqrt<3>1000 1000^\tfrac1310

These examples help us version a relationship between radicals and also rational exponents: namely, that the n^th root the a number can be written as either \sqrtx or x^\frac1n.

Radical Form

Exponent Form

\sqrtx x^\tfrac12
\sqrt<3>x x^\tfrac13
\sqrt<4>x x^\tfrac14
\sqrtx x^\tfrac1n

In the table above, notification how the denominator of the rational exponent determines the index of the root. So, an exponent that \frac12 translates to the square root, an exponent of \frac15 translates to the fifth root or \,^5\hspace-0.1in \sqrt\,\,\,, and \frac18 equates to the eighth source or \,^8\hspace-0.1in \sqrt\,\,\,.


Rewrite the expression through the fractional exponent as a radical. The denominator that the fraction determines the root, in this instance the cube root.

\sqrt<3>2x

The parentheses in \left( 2x \right)^\frac13 show that the exponent refers to everything within the parentheses.

Answer

(2x)^^\frac13=\sqrt<3>2x


Remember that exponents just refer come the quantity automatically to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one necessary difference—there space no parentheses! watch what happens.


Rewrite the expression through the fractional exponent together a radical. The denominator that the portion determines the root, in this situation the cube root.

2\sqrt<3>x

The exponent refers just to the component of the expression automatically to the left that the exponent, in this case x, however not the 2.

Answer

2x^^\frac13=2\sqrt<3>x


Flexibility


We can write radicals with rational exponents, and also as we will see once we simplify more complex radical expressions, this deserve to make things easier. Having various ways come express and write algebraic expressions allows us come have versatility in solving and also simplifying them. The is like having actually a thesaurus once you write, you desire to have options for express yourself!


The radical form \,^4\hspace-0.1in\sqrt\,\,\,\, have the right to be rewritten as the exponent \frac14. Eliminate the radical and also place the exponent beside the base.

81^\frac14

Answer

\sqrt<4>81=81^\frac14


(9.2.4) – rational exponents whose molecule is not equal to one

All of the numerators for the fractional exponents in the examples above were 1. You deserve to use fractional index number that have actually numerators various other than 1 come express roots, as displayed below.

Radical

Exponent

\sqrt99^\frac12
\sqrt<3>9^29^\frac23
\sqrt<4>9^39^\frac34
\sqrt<5>9^29^\frac25
\sqrt9^x9\fracxn


To rewrite a radical utilizing a spring exponent, the power to which the radicand is elevated becomes the numerator and the root/ table of contents becomes the denominator.


Writing Rational Exponents

Any radical in the kind \large \sqrta^m deserve to be written making use of a fountain exponent in the kind \large a^\fracmn.


The relationship between \sqrta^mand a^\fracmn functions for rational exponents that have actually a molecule of 1 as well. For example, the radical \sqrt<3>8 can also be created as \sqrt<3>8^1, since any type of number remains the same value if it is increased to the an initial power. You deserve to now watch where the molecule of 1 originates from in the equivalent type of 8^\frac13.

In the following example, we exercise writing radicals through rational exponents wherein the numerator is not equal to one.


Example

Rewrite the radicals making use of a rational exponent, then simplify your result.

\sqrt<3>a^6\sqrt<12>16^3

1.\sqrta^m deserve to be rewritten as a^\fracmn, so in this situation n=3,\text and m=6, therefore

\sqrt<3>a^6=a^\frac63

Simplify the exponent.

a^\frac63=a^2

Answer

\sqrt<3>a^6=a^2

2. \sqrta^m have the right to be rewritten as a^\fracmn, for this reason in this situation n=12,\text and m=3, therefore

\sqrt<12>16^3=16^\frac312=16^\frac14

Simplify the expression making use of rules for exponents.

\beginarrayccc16=2^4\\16^\frac14=2^4^\frac14\\=2^4\cdot\frac14\\=2^1=2\endarray

Answer

\sqrt<12>16^3=2


In our last instance we will rewrite expressions with rational exponents together radicals. This practice will help us when we simplify more facility radical expressions, and as us learn just how to fix radical equations. Generally it is easier to simplify when we use rational exponents, yet this exercise is to plan to aid you know how the numerator and also denominator that the exponent are the exponent of a radicand and index the a radical.


Example

Rewrite the expressions using a radical.

x^\frac235^\frac47
x^\frac23, the numerator is 2 and the denominator is 3, because of this we will have actually the 3rd root that x squared, \sqrt<3>x^25^\frac47, the numerator is 4 and also the denominator is 7, so we will have actually the seventh root of 5 raised to the 4th power. \sqrt<7>5^4

In the following video we show more examples of composing radical expressions through rational exponents and also expressions v rational exponents as radical expressions.

We will use this notation later, therefore come earlier for exercise if friend forget how to write a radical with a reasonable exponent.

(9.2.5) – simplify Radical Expressions

Radical expressions space expressions that contain radicals. Radical expressions come in many forms, from basic and familiar, such as \sqrt16, to rather complicated, together in \sqrt<3>250x^4y.

To simplify complicated radical expressions, we can use part definitions and also rules from simple exponents. Remind the Product raised to a power Rule from once you learned exponents. This ascendancy states that the product of two or more non-zero numbers raised to a power is same to the product of each number elevated to the exact same power. In math terms, that is composed \left(ab\right)^x=a^x\cdotb^x. So, because that example, you can use the dominion to rewrite \left( 3x \right)^2 as 3^2\cdot x^2=9\cdot x^2=9x^2.

Now rather of utilizing the exponent 2, let’s use the exponent \frac12. The exponent is distributed in the exact same way.

\left( 3x \right)^\frac12=3^\frac12\cdot x^\frac12

And due to the fact that you recognize that increasing a number to the \frac12 strength is the exact same as taking the square source of that number, you can also write the this way.

\sqrt3x=\sqrt3\cdot \sqrtx

Look in ~ that—you have the right to think of any kind of number underneath a radical as the product of different factors, every underneath its own radical.


A Product raised to a power Rule or periodically called The Square source of a Product Rule

For any real number a and b, \sqrtab=\sqrta\cdot \sqrtb.

For example: \sqrt100=\sqrt10\cdot \sqrt10, and \sqrt75=\sqrt25\cdot \sqrt3


This dominion is important because it helps you think the one radical together the product of many radicals. If you deserve to identify perfect squares in ~ a radical, similar to \sqrt(2\cdot 2)(2\cdot 2)(3\cdot 3), you can rewrite the expression together the product of lot of perfect squares: \sqrt2^2\cdot \sqrt2^2\cdot \sqrt3^2.

The square source of a product ascendancy will aid us leveling roots that aren’t perfect, as is shown the complying with example.

Simplify radical expressions making use of factoring
63 is not a perfect square so we can use the square source of a product dominion to simplify any kind of factors that space perfect squares.Factor 63 right into 7 and 9.

\sqrt7\cdot 9

9 is a perfect square, 9=3^2, thus we can rewrite the radicand.

\sqrt7\cdot 3^2

Using the Product raised to a strength rule, different the radical into the product of two factors, each under a radical.

\sqrt7\cdot \sqrt3^2

Take the square source of 3^2.

\sqrt7\cdot 3

Rearrange determinants so the integer shows up before the radical, and also then multiply. (This is done so the it is clean that just the 7 is under the radical, no the 3.)

3\cdot \sqrt7

Answer \sqrt63=3\sqrt7
The final answer 3\sqrt7 might look a little odd, but it is in simplified form. You deserve to read this as “three radical seven” or “three time the square root of seven.”

The following video shows more examples of just how to simplify square root that do not have actually perfect square radicands.

Before we move on to simple more complex radicals with variables, we need to discover about an essential behavior the square roots through variables in the radicand.

Consider the expression \sqrtx^2. This looks like it need to be same to x, right? Let’s test some worths for x and also see what happens.

In the chart below, look along each row and also determine even if it is the value of x is the same as the value of \sqrtx^2. Where space they equal? Where are they no equal?

After doing that for every row, watch again and also determine whether the worth of \sqrtx^2 is the same as the value of \left|x\right|.

xx^2\sqrtx^2\left|x\right|
−52555
−2422
0000
63666
101001010

Notice—in cases where x is a an adverse number, \sqrtx^2\neqx! However, in all instances \sqrtx^2=\left|x\right|. You require to take into consideration this reality when simplifying radicals v an even index the contain variables, due to the fact that by definition \sqrtx^2 is constantly nonnegative.


Taking the Square source of a Radical Expression

When finding the square source of an expression that consists of variables increased to a power, consider that \sqrtx^2=\left|x\right|.

Examples: \sqrt9x^2=3\left|x\right|, and also \sqrt16x^2y^2=4\left|xy\right|


We will incorporate this v the square source of a product rule in our next instance to leveling an expression with three variables in the radicand.


Factor to discover variables with even exponents.

\sqrta^2\cdot a\cdot b^4\cdot b\cdot c^2

Rewrite b^4 as \left(b^2\right)^2.

\sqrta^2\cdot a\cdot (b^2)^2\cdot b\cdot c^2

Separate the squared components into separation, personal, instance radicals.

\sqrta^2\cdot \sqrt(b^2)^2\cdot \sqrtc^2\cdot \sqrta\cdot b

Take the square source of each radical. Remember the \sqrta^2=\left| a \right|.

\left| a \right|\cdot b^2\cdot \left|c\right|\cdot \sqrta\cdot b

Simplify and multiply.

\left| ac \right|b^2\sqrtab

Answer

\sqrta^3b^5c^2=\left| ac \right|b^2\sqrtab

Analysis the the Solution

Why didn’t we write b^2 as |b^2|? since when friend square a number, friend will always get a confident result, therefore the principal square source of \left(b^2\right)^2 will constantly be non-negative. One pointer for knowing once to use the absolute worth after simplifying any type of even indexed root is come look at the last exponent on her variable terms. If the exponent is strange – including 1 – include an pure value. This uses to simplifying any type of root through an even index, as we will check out in later examples.


In the following video clip you will certainly see an ext examples of just how to leveling radical expressions through variables.

We will certainly show an additional example whereby the simplified expression contains variables through both odd and also even powers.


Factor to uncover identical pairs.

\sqrt3\cdot 3\cdot x^3\cdot x^3\cdot y^2\cdot y^2

Rewrite the pairs together perfect squares.

\sqrt3^2\cdot \left( x^3 \right)^2\cdot \left( y^2 \right)^2

Separate into individual radicals.

\sqrt3^2\cdot \sqrt\left( x^3 \right)^2\cdot \sqrt\left( y^2 \right)^2

Simplify.

3x^3y^2

Because x has an weird power, we will include the absolute value for our final solution.

3|x^3|y^2

Answer

\sqrt9x^6y^4=3|x^3|y


First, we aspect the radicand:

x^2-6x+9 = (x-3)^2

Then, we rewrite the radical expression and take the square root:

\sqrtx^2-6x+9=\sqrt(x-3)^2 = |x-3|

Answer

|x-3|

Analysis the the solution:

Note, that if we didn’t include the absolute worth signs, the two sides the the equation would be various for values of x less than 3. For example, assessing the radical expression at x=1 would offer us \sqrt(1-3)^2=\sqrt(-2)^2=2; and plugging in x=1 right into our last answer, additionally yields: |1-3|=2. However, if us did not put absolute value signs, plugging in x=1 right into x-3 would yield 1-3=-2, a different value.


Look because that squared numbers and variables. Element 49 right into 7\cdot7, x^10 into x^5\cdotx^5, and y^8 into y^4\cdoty^4.

\sqrt7\cdot 7\cdot x^5\cdot x^5\cdot y^4\cdot y^4

Rewrite the pairs as squares.

\sqrt7^2\cdot (x^5)^2\cdot (y^4)^2

Separate the squared determinants into individual radicals.

\sqrt7^2\cdot \sqrt(x^5)^2\cdot \sqrt(y^4)^2

Take the square root of each radical using the ascendancy that \sqrtx^2=x.

7\cdot x^5\cdot y^4

Multiply.

7x^5y^4

Answer

\sqrt49x^10y^8=7|x^5|y^4


Factor 40 into prime factors.

\sqrt<3>5\cdot 2\cdot 2\cdot 2\cdot m^5

Since girlfriend are in search of the cube root, you must find components that appear 3 time under the radical. Rewrite 2\cdot 2\cdot 2 together 2^3.

\sqrt<3>2^3\cdot 5\cdot m^5

Rewrite m^5 together m^3\cdot m^2.

\sqrt<3>2^3\cdot 5\cdot m^3\cdot m^2

Rewrite the expression together a product of multiple radicals.

\sqrt<3>2^3\cdot \sqrt<3>5\cdot \sqrt<3>m^3\cdot \sqrt<3>m^2

Simplify and multiply.

2\cdot \sqrt<3>5\cdot m\cdot \sqrt<3>m^2

Answer

\sqrt<3>40m^5=2m\sqrt<3>5m^2


Factor the expression right into cubes.

Separate the cubed determinants into individual radicals.

\beginarrayr\sqrt<3>-1\cdot 27\cdot x^4\cdot y^3 &=& \sqrt<3>(-1)^3\cdot (3)^3\cdot x^3\cdot x\cdot y^3\\ &=& \sqrt<3>(-1)^3\cdot \sqrt<3>(3)^3\cdot \sqrt<3>x^3\cdot \sqrt<3>x\cdot \sqrt<3>y^3\endarray

Simplify the cube roots.

-1\cdot 3\cdot x\cdot y\cdot \sqrt<3>x

Answer

\sqrt<3>-27x^4y^3=-3xy\sqrt<3>x

Analysis that the solution

You could examine your prize by performing the train station operation. If you are right, once you cube -3xy\sqrt<3>x friend should gain -27x^4y^3.

\beginarrayl\left( -3xy\sqrt<3>x \right)\left( -3xy\sqrt<3>x \right)\left( -3xy\sqrt<3>x \right) &=& -3\cdot -3\cdot -3\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y\cdot \sqrt<3>x\cdot \sqrt<3>x\cdot \sqrt<3>x\\ &=& -27\cdot x^3\cdot y^3\cdot \sqrt<3>x^3\\ &=& -27x^3y^3\cdot x\\ &=& -27x^4y^3\endarray


Factor −24 to uncover perfect cubes. Here, −1 and also 8 room the perfect cubes.

\sqrt<3>-1\cdot 8\cdot 3\cdot a^5

Factor variables. You are looking for cube exponents, so you factor a^5 into a^3 and a^2.

\sqrt<3>(-1)^3\cdot 2^3\cdot 3\cdot a^3\cdot a^2

Separate the factors into individual radicals.

\sqrt<3>(-1)^3\cdot \sqrt<3>2^3\cdot \sqrt<3>a^3\cdot \sqrt<3>3\cdot a^2

Simplify, making use of the property \sqrt<3>x^3=x. 

-1\cdot 2\cdot a\cdot \sqrt<3>3\cdot a^2

This is the simplest type of this expression; every cubes have actually been pulled out of the radical expression.

-2a\sqrt<3>3a^2

Answer

\sqrt<3>-24a^5=-2a\sqrt<3>3a^2

Analysis of the solution

You can check your answer by squaring it to be sure it equals 100x^2y^4.


Rewrite the radical using rational exponents.

(81x^8y^3)^\frac14

Use the rules of index number to leveling the expression.

\beginarrayr81^\frac14\cdot x^\frac84\cdot y^\frac34&=& (3\cdot 3\cdot 3\cdot 3)^\frac14x^2y^\frac34\\ &=& (3^4)^\frac14x^2y^\frac34\\ &=&3x^2y^\frac34\endarray

Change the expression with the rational exponent back to radical form.

3x^2\sqrt<4>y^3

Answer

\sqrt<4>81x^8y^3=3x^2\sqrt<4>y^3


In the following video clip we show one more example of how to leveling a fourth and also fifth root.

For our last example, we will certainly simplify a more complex expression, \large\frac10b^2c^2c\sqrt<3>8b^4. This expression has two variables, a fraction, and a radical. Let’s take it step-by-step and also see if utilizing fractional exponents can assist us simplify it.We will start by simple the denominator, due to the fact that this is wherein the radical sign is located. Recall that an exponent in the denominator or a portion can be rewritten together a an adverse exponent.


Separate the determinants in the denominator.

\displaystyle \frac10b^2c^2c\cdot \sqrt<3>8\cdot \sqrt<3>b^4

Take the cube source of 8, i m sorry is 2.

\displaystyle \frac10b^2c^2c\cdot 2\cdot \sqrt<3>b^4

Rewrite the radical utilizing a spring exponent.

\displaystyle \frac10b^2c^2c\cdot 2\cdot b^\frac43

Rewrite the fraction as a series of components in order to cancel factors (see following step).

\displaystyle \frac102\cdot \fracc^2c\cdot \fracb^2b^\frac43

Simplify the consistent and c factors.

\displaystyle 5\cdot c\cdot \fracb^2b^\frac43

Use the preeminence of an adverse exponents, \displaystyle n^-x = \frac1n^x, to rewrite \displaystyle \frac1b^\tfrac43 as \displaystyle b^-\tfrac43.

\displaystyle 5cb^2b^-\ \frac43

Combine the b determinants by adding the exponents.

\displaystyle 5cb^\frac23

Change the expression with the fractional exponent ago to radical form. By convention, one expression is no usually considered simplified if it has actually a fractional exponent or a radical in the denominator.

5c\sqrt<3>b^2

Answer

\displaystyle \frac10b^2c^2c\sqrt<3>8b^4=5c\sqrt<3>b^2


Well, the took a while, yet you walk it. You used what friend know about fractional exponents, an adverse exponents, and also the rule of exponents to simplify the expression.

In our last video we display how to use rational exponents to leveling radical expressions.

Summary

A radical expression is a mathematical way of representing the nth root of a number. Square roots and also cube roots room the most common radicals, however a root have the right to be any type of number. To simplify radical expressions, look for exponential components within the radical, and then use the building \sqrtx^n=x if n is odd, and also \sqrtx^n=\left| x \right| if n is even to pull the end quantities. All rules of essence operations and also exponents apply when simple radical expressions.

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The steps to consider when simple a radical room outlined below.


Simplifying a radical

When working with exponents and also radicals:

If n is odd, \sqrtx^n=x.If n is even, \sqrtx^n=\left| x \right|. (The absolute value accounts because that the fact that if x is an adverse and elevated to an also power, the number will certainly be positive, as will certainly the nth principal root of the number.)