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Nostrana [21]
3 years ago
10

Find the GCF of

" class="latex-formula"> and 30x^{5}
90x(3)

6x(3)

90x^{5}

6x(5)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The numbers in parentheses are exponents.
Mathematics
2 answers:
melamori03 [73]3 years ago
8 0

18x^3 = 6x^3 *  1

30x^5 = 6x^3 * 5x^2


So GCF = 6x^3


Answer

6x^3

GarryVolchara [31]3 years ago
3 0

18x^3=\boxed{2}\cdot\boxed{3}\cdot3\cdot \boxed{x}\cdot\boxed{x}\cdot\boxed{x}\\\\30x^5=\boxed{2}\cdot\boxed{3}\cdot5\cdot\boxed{x}\cdot\boxed{x}\cdot\boxed{x}\cdot x\cdot x\\\\GCF(18x^3;\ 30x^5)=\boxed{2}\cdot\boxed{3}\cdot\boxed{x}\cdot\boxed{x}\cdot\boxed{x}=\boxed{6x^3}

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Answer:

\displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1

General Formulas and Concepts:

<u>Pre-Algebra</u>

  • Order of Operations
  • Equality Properties

<u>Algebra I</u>

  • Functions
  • Function Notation
  • Exponential Rule [Rewrite]:                                                                              \displaystyle b^{-m} = \frac{1}{b^m}

<u>Algebra II</u>

  • Natural logarithms ln and Euler's number e

<u>Calculus</u>

Derivatives

Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Slope Fields

  • Separation of Variables
  • Solving Differentials

Integrals

  • Antiderivatives

Integration Constant C

Integration Rule [Reverse Power Rule]:                                                                   \displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C

Integration Property [Addition/Subtraction]:                                                           \displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx

U-Substitution

Logarithmic Integration:                                                                                            \displaystyle \int {\frac{1}{u}} \, dx = ln|u| + C

Step-by-step explanation:

*Note:  

When solving differential equations in slope fields, disregard the integration constant C for variable y.

<u />

<u>Step 1: Define</u>

\displaystyle \frac{dy}{dx} = x^2(y - 1)

\displaystyle f(0) = 3

<u>Step 2: Rewrite</u>

<em>Separation of Variables. Get differential equation to a form where we can integrate both sides and rewrite Leibniz Notation.</em>

  1. [Separation of Variables] Rewrite Leibniz Notation:                                      \displaystyle dy = x^2(y - 1) \ dx
  2. [Separation of Variables] Isolate <em>y</em>'s together:                                               \displaystyle \frac{1}{y - 1} \ dy = x^2 \ dx

<u>Step 3: Find General Solution Pt. 1</u>

  1. [Differential] Integrate both sides:                                                                   \displaystyle \int {\frac{1}{y - 1}} \, dy = \int {x^2} \, dx
  2. [dx Integral] Integrate [Integration Rule - Reverse Power Rule]:                   \displaystyle \int {\frac{1}{y - 1}} \, dy = \frac{x^3}{3} + C

<u>Step 4: Find General Solution Pt. 2</u>

<em>Identify variables for u-substitution for dy.</em>

  1. Set:                                                                                                                    \displaystyle u = y - 1
  2. Differentiate [Basic Power Rule]:                                                                     \displaystyle du = dy

<u>Step 5: Find General Solution Pt. 3</u>

  1. [dy Integral] U-Substitution:                                                                             \displaystyle \int {\frac{1}{u}} \, du = \frac{x^3}{3} + C
  2. [dy Integral] Integrate [Logarithmic Integration]:                                            \displaystyle ln|u| = \frac{x^3}{3} + C
  3. [Equality Property] e both sides:                                                                     \displaystyle e^\bigg{ln|u|} = e^\bigg{\frac{x^3}{3} + C}
  4. Simplify:                                                                                                             \displaystyle |u| = Ce^\bigg{\frac{x^3}{3}}
  5. Rewrite:                                                                                                             \displaystyle u = \pm Ce^\bigg{\frac{x^3}{3}}
  6. Back-Substitute:                                                                                               \displaystyle y - 1 = \pm Ce^\bigg{\frac{x^3}{3}}
  7. [Equality Property] Isolate <em>y</em>:                                                                            \displaystyle y = \pm Ce^\bigg{\frac{x^3}{3}} + 1

General Form:  \displaystyle y = \pm Ce^\bigg{\frac{x^3}{3}} + 1

<u>Step 6: Find Particular Solution</u>

  1. Substitute in function values [General Form]:                                                \displaystyle 3 = \pm Ce^\bigg{\frac{0^3}{3}} + 1
  2. Simplify:                                                                                                             \displaystyle 3 = \pm C + 1
  3. [Equality Property] Isolate <em>C</em>:                                                                           \displaystyle 2 = \pm C
  4. Rewrite:                                                                                                             \displaystyle C = 2
  5. Substitute in <em>C</em> [General Form]:                                                                       \displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1

∴ our particular solution is  \displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentials and Slope Fields

Book: College Calculus 10e  

6 0
3 years ago
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