Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Complete this item. (Put all variable answers in alphabetical order.) Simplify: (39x2 18x²y Xyz)( x++ E012 3 4|56|7 8|9| || |3 F F NEXT QUESTION O ASK FOR HELP TURN IT IN O 2007, 2009, 2011, 2012, 2013, 2014, 2016 Glynlyon, Inc. TERMS OF USE
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![### Simplifying Algebraic Expressions
To simplify an algebraic expression using multiplication and exponents, follow these steps. The example provided is:
\[ \text{Simplify:} \quad (3x)^3 z^4 \cdot 18x^2 y \]
**Step-by-Step Solution:**
1. **Distribute the Exponents:**
Distribute the exponent in \((3x)^3\):
\[
(3x)^3 = 3^3 \cdot x^3 = 27x^3
\]
2. **Break Down the Multiplication:**
The expression now looks like:
\[
27x^3 z^4 \cdot 18x^2 y
\]
3. **Multiply the Coefficients:**
Multiply the numerical coefficients \(27\) and \(18\):
\[
27 \cdot 18 = 486
\]
4. **Combine Like Terms with Exponents:**
- For \(x\):
\[
x^3 \cdot x^2 = x^{3+2} = x^5
\]
- Other variables \(z^4\) and \(y\) do not change:
\[
z^4 \quad \text{and} \quad y
\]
5. **Construct the Final Expression:**
Combine all parts to form the simplified expression:
\[
486x^5 y z^4
\]
**Result:**
\[
486x^5 y z^4
\]
**Interactive Component:**
When you use the interactive tool on the website:
- **Click an item** from the list or group of pictures at the bottom of the problem and drag it into the correct position in the answer box.
- **Release your mouse button** when the item is placed.
- If you need to change your mind, drag the item to the trashcan.
- **Click the trashcan** to clear all your answers.
The following image depicts the drag-and-drop feature used to input your answer in alphabetical order. Here, the variable ‘y’ is placed in the answer box to indicate that the complete answer includes variable terms arranged alphabetically.
**Tags:**
- Variables: \(x, y, z\)
- Exponents
- Algebra
**User Interface Description](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefebba2e-ab3a-49b5-a7d3-0c6a95c353e0%2F084e5f22-334b-4526-9ae1-698ba05bebc1%2Fid5gl96_processed.jpeg&w=3840&q=75)
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