Simplify The Expression 49 1/2
Learning Objectives
- (ix.ii.ane) – Define and place a radical expression
- (9.2.2) – Convert radicals to expressions with rational exponents
- (9.2.3) – Convert expressions with rational exponents to their radical equivalent
- (9.2.4) – Rational exponents whose numerator is not equal to one
- (9.2.v) – Simplify Radical Expressions
- Simplify radical expressions using factoring
- Simplify radical expressions using rational exponents and the laws of exponents
(ix.two.ane) – Ascertain and identify a radical expression
Square roots are most often written using a radical sign, like this, [latex] \sqrt{4}[/latex]. But at that place is some other way to represent them. Y'all can use rational exponents instead of a radical. A rational exponent is an exponent that is a fraction. For example, [latex] \sqrt{four}[/latex] can be written as [latex] {{4}^{\tfrac{ane}{two}}}[/latex].
Tin't imagine raising a number to a rational exponent? They may exist hard to get used to, but rational exponents can actually help simplify some problems. Writing radicals with rational exponents will come in handy when we talk over techniques for simplifying more complex radical expressions.
Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from elementary and familiar, such as[latex] \sqrt{16}[/latex], to quite complicated, as in [latex] \sqrt[3]{250{{ten}^{4}}y}[/latex]
(nine.2.2) – Convert radicals to expressions with rational exponents
Radicals and fractional exponents are alternate means of expressing the same affair. In the tabular array below we show equivalent ways to express radicals: with a root, with a rational exponent, and every bit a principal root.
| Radical Form | Exponent Form | Main Root |
|---|---|---|
| [latex] \sqrt{16}[/latex] | [latex] {{16}^{\tfrac{1}{2}}}[/latex] | four |
| [latex] \sqrt{25}[/latex] | [latex] {{25}^{\tfrac{ane}{ii}}}[/latex] | 5 |
| [latex] \sqrt{100}[/latex] | [latex] {{100}^{\tfrac{ane}{2}}}[/latex] | ten |
Let's expect at some more examples, but this fourth dimension with cube roots. Recollect, cubing a number raises it to the power of three. Notice that in the examples in the table below, the denominator of the rational exponent is the number three.
| Radical Grade | Exponent Form | Main Root |
|---|---|---|
| [latex] \sqrt[iii]{eight}[/latex] | [latex] {{viii}^{\tfrac{one}{iii}}}[/latex] | 2 |
| [latex] \sqrt[three]{eight}[/latex] | [latex] {{125}^{\tfrac{one}{3}}}[/latex] | 5 |
| [latex] \sqrt[3]{1000}[/latex] | [latex] {{1000}^{\tfrac{1}{3}}}[/latex] | 10 |
These examples assistance us model a relationship between radicals and rational exponents: namely, that the [latex]n^{thursday}[/latex] root of a number tin can exist written equally either [latex] \sqrt[n]{10}[/latex] or [latex] {{x}^{\frac{one}{north}}}[/latex].
| Radical Form | Exponent Form |
|---|---|
| [latex] \sqrt{x}[/latex] | [latex] {{ten}^{\tfrac{1}{2}}}[/latex] |
| [latex] \sqrt[3]{x}[/latex] | [latex] {{x}^{\tfrac{1}{3}}}[/latex] |
| [latex] \sqrt[4]{10}[/latex] | [latex] {{10}^{\tfrac{1}{four}}}[/latex] |
| … | … |
| [latex] \sqrt[n]{ten}[/latex] | [latex] {{x}^{\tfrac{one}{n}}}[/latex] |
In the table above, notice how the denominator of the rational exponent determines the alphabetize of the root. So, an exponent of [latex] \frac{1}{2}[/latex] translates to the square root, an exponent of [latex] \frac{ane}{5}[/latex] translates to the 5th root or [latex]{\,}^5\hspace{-0.1in} \sqrt{\,\,\,}[/latex], and [latex] \frac{1}{8}[/latex] translates to the eighth root or [latex]{\,}^8\hspace{-0.1in} \sqrt{\,\,\,}[/latex].
Example
Express [latex] {{(2x)}^{^{\frac{ane}{iii}}}}[/latex] in radical form.
Call back that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one of import difference—there are no parentheses! Expect what happens.
Example
Limited [latex] 2{{ten}^{^{\frac{1}{iii}}}}[/latex] in radical form.
(9.two.iii) – Convert expressions with rational exponents to their radical equivalent
Flexibility
We tin write radicals with rational exponents, and as nosotros will see when we simplify more complex radical expressions, this can make things easier. Having different ways to express and write algebraic expressions allows the states to take flexibility in solving and simplifying them. It is similar having a thesaurus when you write, you want to take options for expressing yourself!
Example
Write [latex] \sqrt[4]{81}[/latex] as an expression with a rational exponent.
Instance
Limited [latex] 4\sqrt[3]{xy}[/latex] with rational exponents.
(ix.ii.iv) – Rational exponents whose numerator is not equal to i
All of the numerators for the partial exponents in the examples above were 1. Y'all tin can apply partial exponents that have numerators other than one to express roots, as shown below.
| Radical | Exponent |
|---|---|
| [latex] \sqrt{nine}[/latex] | [latex]9^{\frac{1}{2}}[/latex] |
| [latex] \sqrt[3]{{{ix}^{2}}}[/latex] | [latex]nine^{\frac{2}{three}}[/latex] |
| [latex]\sqrt[4]{nine^{3}}[/latex] | [latex]9^{\frac{3}{4}}[/latex] |
| [latex]\sqrt[five]{9^{ii}}[/latex] | [latex]nine^{\frac{2}{five}}[/latex] |
| … | … |
| [latex]\sqrt[n]{9^{x}}[/latex] | [latex]9\frac{10}{n}[/latex] |
To rewrite a radical using a fractional exponent, the power to which the radicand is raised becomes the numerator and the root/ index becomes the denominator.
Writing Rational Exponents
Whatever radical in the form [latex]\large \sqrt[n]{a^{k}}[/latex] can be written using a fractional exponent in the class [latex]\large a^{\frac{m}{n}}[/latex].
The relationship between [latex] \sqrt[n]{{{a}^{m}}}[/latex]and [latex] {{a}^{\frac{m}{n}}}[/latex] works for rational exponents that have a numerator of 1 as well. For example, the radical [latex] \sqrt[3]{viii}[/latex] can likewise be written as [latex] \sqrt[3]{{{8}^{1}}}[/latex], since any number remains the aforementioned value if it is raised to the offset power. You can now see where the numerator of ane comes from in the equivalent class of [latex] {{viii}^{\frac{1}{3}}}[/latex].
In the side by side instance, we practice writing radicals with rational exponents where the numerator is not equal to one.
Example
Rewrite the radicals using a rational exponent, so simplify your result.
- [latex]\sqrt[3]{{{a}^{half-dozen}}}[/latex]
- [latex]\sqrt[12]{16^3}[/latex]
In our last example we will rewrite expressions with rational exponents as radicals. This practice will assistance us when we simplify more complicated radical expressions, and as we learn how to solve radical equations. Typically information technology is easier to simplify when we utilize rational exponents, but this practise is intended to help you empathise how the numerator and denominator of the exponent are the exponent of a radicand and index of a radical.
Example
Rewrite the expressions using a radical.
- [latex]{x}^{\frac{2}{3}}[/latex]
- [latex]{5}^{\frac{four}{7}}[/latex]
In the post-obit video we bear witness more examples of writing radical expressions with rational exponents and expressions with rational exponents every bit radical expressions.
We volition use this note later, so come up dorsum for do if y'all forget how to write a radical with a rational exponent.
(9.2.five) – Simplify Radical Expressions
Radical expressions are expressions that comprise radicals. Radical expressions come in many forms, from unproblematic and familiar, such every bit[latex] \sqrt{16}[/latex], to quite complicated, as in [latex] \sqrt[3]{250{{ten}^{four}}y}[/latex].
To simplify complicated radical expressions, we can utilise some definitions and rules from simplifying exponents. Think the Product Raised to a Power Rule from when y'all studied exponents. This dominion states that the product of 2 or more not-nothing numbers raised to a ability is equal to the product of each number raised to the same power. In math terms, it is written [latex]\left(ab\right)^{x}=a^{10}\cdot{b}^{10}[/latex]. So, for example, you lot tin can use the rule to rewrite [latex] {{\left( 3x \correct)}^{2}}[/latex] as [latex] {{3}^{2}}\cdot {{x}^{2}}=9\cdot {{x}^{2}}=nine{{x}^{2}}[/latex].
Now instead of using the exponent 2, allow'due south apply the exponent [latex] \frac{ane}{2}[/latex]. The exponent is distributed in the same manner.
[latex] {{\left( 3x \right)}^{\frac{1}{two}}}={{3}^{\frac{i}{2}}}\cdot {{ten}^{\frac{1}{2}}}[/latex]
And since you know that raising a number to the [latex] \frac{1}{2}[/latex] power is the same as taking the square root of that number, you can also write it this way.
[latex] \sqrt{3x}=\sqrt{three}\cdot \sqrt{x}[/latex]
Look at that—yous can think of any number underneath a radical as the product of separate factors, each underneath its own radical.
A Product Raised to a Power Rule or sometimes called The Square Root of a Product Rule
For any existent numbers [latex]a[/latex] and [latex]b[/latex], [latex] \sqrt{ab}=\sqrt{a}\cdot \sqrt{b}[/latex].
For example: [latex] \sqrt{100}=\sqrt{ten}\cdot \sqrt{ten}[/latex], and [latex] \sqrt{75}=\sqrt{25}\cdot \sqrt{iii}[/latex]
This dominion is important because it helps you think of ane radical as the production of multiple radicals. If you can identify perfect squares within a radical, as with [latex] \sqrt{(2\cdot 2)(2\cdot 2)(iii\cdot 3})[/latex], you can rewrite the expression as the production of multiple perfect squares: [latex] \sqrt{{{2}^{two}}}\cdot \sqrt{{{2}^{2}}}\cdot \sqrt{{{3}^{2}}}[/latex].
The square root of a product rule will aid u.s. simplify roots that aren't perfect, equally is shown the following instance.
Simplify radical expressions using factoring
Instance
Simplify. [latex] \sqrt{63}[/latex]
The last answer [latex] iii\sqrt{7}[/latex] may look a fleck odd, but information technology is in simplified grade. You tin read this every bit "three radical seven" or "3 times the foursquare root of seven."
The following video shows more examples of how to simplify square roots that do not have perfect foursquare radicands.
Earlier nosotros move on to simplifying more complex radicals with variables, we need to learn about an of import behavior of square roots with variables in the radicand.
Consider the expression [latex] \sqrt{{{x}^{two}}}[/latex]. This looks like it should be equal to x, correct? Allow'southward test some values for x and see what happens.
In the chart beneath, look along each row and make up one's mind whether the value of x is the aforementioned every bit the value of [latex] \sqrt{{{x}^{two}}}[/latex]. Where are they equal? Where are they not equal?
After doing that for each row, expect again and determine whether the value of [latex] \sqrt{{{x}^{ii}}}[/latex] is the aforementioned as the value of [latex]\left|ten\right|[/latex].
| [latex]x[/latex] | [latex]x^{2}[/latex] | [latex]\sqrt{x^{2}}[/latex] | [latex]\left|x\right|[/latex] |
|---|---|---|---|
| [latex]−5[/latex] | 25 | five | v |
| [latex]−2[/latex] | 4 | two | 2 |
| 0 | 0 | 0 | 0 |
| vi | 36 | 6 | 6 |
| 10 | 100 | 10 | 10 |
Discover—in cases where x is a negative number, [latex]\sqrt{ten^{two}}\neq{10}[/latex]! Yet, in all cases [latex]\sqrt{x^{2}}=\left|x\right|[/latex]. You need to consider this fact when simplifying radicals with an even index that contain variables, considering by definition [latex]\sqrt{10^{2}}[/latex] is always nonnegative.
Taking the Square Root of a Radical Expression
When finding the square root of an expression that contains variables raised to a power, consider that [latex]\sqrt{x^{2}}=\left|x\right|[/latex].
Examples: [latex]\sqrt{9x^{two}}=iii\left|x\right|[/latex], and [latex]\sqrt{16{{x}^{ii}}{{y}^{2}}}=4\left|xy\right|[/latex]
We will combine this with the square root of a product rule in our adjacent example to simplify an expression with three variables in the radicand.
Example
Simplify. [latex] \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[/latex]
In the post-obit video you volition meet more than examples of how to simplify radical expressions with variables.
We will bear witness another instance where the simplified expression contains variables with both odd and even powers.
Case
Simplify. [latex] \sqrt{9{{x}^{half-dozen}}{{y}^{4}}}[/latex]
Example
Simplify [latex]\sqrt{x^2-6x+9}[/latex].
In our adjacent instance we will start with an expression written with a rational exponent. You volition see that yous tin use a similar process – factoring and sorting terms into squares – to simplify this expression.
Example
Simplify. [latex] {{(36{{x}^{four}})}^{\frac{one}{two}}}[/latex]
Hither is one more example with perfect squares.
Case
Simplify. [latex] \sqrt{49{{ten}^{x}}{{y}^{8}}}[/latex]
Simplify cube roots
Nosotros can use the aforementioned techniques nosotros accept used for simplifying square roots to simplify higher order roots. For example to simplify a cube root, the goal is to find factors under the radical that are perfect cubes then that you tin have their cube root. We no longer demand to be concerned about whether we have identified the principal root since we are at present finding cube roots. Focus on finding identical trios of factors as you simplify.
Example
Simplify. [latex] \sqrt[iii]{twoscore{{m}^{5}}}[/latex]
Recall that you tin can take the cube root of a negative expression. In the next example we will simplify a cube root with a negative radicand.
Case
Simplify. [latex] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[/latex]
Yous can also skip the step of factoring out the negative one once you are comfy with identifying cubes.
Example
Simplify. [latex] \sqrt[3]{-24{{a}^{five}}}[/latex]
In the following video we testify more examples of simlifying cube roots.
Simplifying 4th roots
Now allow's motility to simplifying fourth caste roots. No affair what root you are simplifying, the same thought applies, find cubes for cube roots, powers of four for fourth roots, etc. Recollect that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to utilize an absolute value.
Case
Simplify. [latex] \sqrt[iv]{81{{x}^{viii}}{{y}^{3}}}[/latex]
Simplify radical expressions using rational exponents and the laws of exponents
An alternative method to factoring is to rewrite the expression with rational exponents, then apply the rules of exponents to simplify. You may find that you prefer one method over the other. Either style, information technology is nice to have options. Nosotros will show the last example once again, using this thought.
Instance
Simplify. [latex] \sqrt[4]{81{{x}^{8}}{{y}^{iii}}}[/latex]
In the following video we show some other instance of how to simplify a 4th and fifth root.
For our last example, we will simplify a more complicated expression, [latex]\big\frac{10{{b}^{ii}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{four}}}}[/latex]. This expression has two variables, a fraction, and a radical. Let's have it step-by-step and see if using fractional exponents can assistance us simplify it.
We will outset past simplifying the denominator, since this is where the radical sign is located. Recall that an exponent in the denominator or a fraction tin exist rewritten as a negative exponent.
Example
Simplify. [latex]\displaystyle \frac{10{{b}^{two}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}[/latex]
Well, that took a while, but you did it. You applied what you lot know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.
In our last video we prove how to use rational exponents to simplify radical expressions.
Summary
A radical expression is a mathematical way of representing the nth root of a number. Square roots and cube roots are the most common radicals, simply a root tin can exist whatsoever number. To simplify radical expressions, look for exponential factors within the radical, and and then use the holding [latex] \sqrt[n]{{{x}^{n}}}=x[/latex] if n is odd, and [latex] \sqrt[n]{{{x}^{northward}}}=\left| x \right|[/latex] if n is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.
The steps to consider when simplifying a radical are outlined below.
Simplifying a radical
When working with exponents and radicals:
- If n is odd, [latex] \sqrt[northward]{{{ten}^{n}}}=x[/latex].
- If n is even, [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex]. (The accented value accounts for the fact that if x is negative and raised to an even power, that number will be positive, as will the nth principal root of that number.)
Simplify The Expression 49 1/2,
Source: https://courses.lumenlearning.com/cuny-hunter-collegealgebra/chapter/1646/
Posted by: kellylithen.blogspot.com
0 Response to "Simplify The Expression 49 1/2"
Post a Comment