Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If f^(')(x)=1//f(x) and f(0)=2, then f(4) can be written in fully simplified form as msqrtn for some integers m and n. What are m and n ? {:[m=],[n=]:}

If f′(x)=1/f(x) f^{\prime}(x)=1 / f(x) and f(0)=2 f(0)=2 , then f(4) f(4) can be written in fully simplified form as mn m \sqrt{n} for some integers m m and n n .\newlineWhat are m m and n n ?\newlinem=□n=□ \begin{array}{l} m= \square \\ n= \square \end{array}

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Negative exponentsright chevron icon

Full solution

Q. If f′(x)=1/f(x) f^{\prime}(x)=1 / f(x) and f(0)=2 f(0)=2 , then f(4) f(4) can be written in fully simplified form as mn m \sqrt{n} for some integers m m and n n .\newlineWhat are m m and n n ?\newlinem=□n=□ \begin{array}{l} m= \square \\ n= \square \end{array}
  1. Identify Given and Asked: Identify the given information and what is asked.\newlineGiven: f′(x)=1f(x)f'(x) = \frac{1}{f(x)} and f(0)=2f(0) = 2.\newlineAsked: f(4)f(4) in the form of mnm\sqrt{n}.
  2. Recognize Exponential Function: Recognize that f′(x)=1f(x)f^{\prime}(x) = \frac{1}{f(x)} suggests that f(x)f(x) is an exponential function.\newlineAssume f(x)=axf(x) = a^{x}, then f′(x)=ln⁡(a)⋅axf^{\prime}(x) = \ln(a) \cdot a^{x}.
  3. Equation Simplification: Since f′(x)=1f(x)f^{'}(x) = \frac{1}{f(x)}, we have ln⁡(a)⋅ax=1ax\ln(a) \cdot a^{x} = \frac{1}{a^{x}}. This implies ln⁡(a)=1a2x\ln(a) = \frac{1}{a^{2x}}.
  4. Find Base from Initial Condition: Use the initial condition f(0)=2f(0) = 2 to find the base aa.f(0)=a0=2f(0) = a^{0} = 2, so a=2a = 2.
  5. Calculate f(4)f(4): Now we know f(x)=2xf(x) = 2^{x}.\newlineTo find f(4)f(4), calculate 242^{4}.\newlinef(4)=24=16f(4) = 2^{4} = 16.
  6. Express in mnm\sqrt{n}: Express 1616 in the form of mnm\sqrt{n}.16=4×4=4×4=4416 = 4 \times 4 = 4 \times \sqrt{4} = 4\sqrt{4}.So, m=4m = 4 and n=4n = 4.

More problems from Negative exponents

Question
Order the expressions from least to greatest.\newline5242+2262−6 5^{2} \quad 4^{2}+2^{2} \quad 6^{2}-6
Get tutor helpright-arrow

Posted 10 months ago

Question
Evaluate the expression. Do not round your answer.\newline15(3+2+5)2= \frac{1}{5}(3+2+5)^{2}=
Get tutor helpright-arrow

Posted 10 months ago

Question
Evaluate the expression.\newlineDo not round your answer.\newline5−322= 5-\frac{3^{2}}{2}=
Get tutor helpright-arrow

Posted 10 months ago

Question
Evaluate the following expression.\newline83−9⋅2÷3= 8^{3}-9 \cdot 2 \div 3=
Get tutor helpright-arrow

Posted 10 months ago

Question
Evaluate the expression. Do not round your answer.\newline14(6+3+1)2= \frac{1}{4}(6+3+1)^{2}=
Get tutor helpright-arrow

Posted 10 months ago

Question
Evaluate the expression. Do not round your answer.\newline(2⋅3)2+52= (2 \cdot 3)^{2}+5^{2}=
Get tutor helpright-arrow

Posted 10 months ago

Question
Evaluate the expression. Do not round your answer.\newline13(4⋅3)+23= \frac{1}{3}(4 \cdot 3)+2^{3}=
Get tutor helpright-arrow

Posted 10 months ago

Question
Evaluate the expression.\newlineDo not round your answer.\newline2(32+42)= 2\left(3^{2}+4^{2}\right)=
Get tutor helpright-arrow

Posted 10 months ago

Question
Evaluate.\newline2×10−7×2= 2 \times 10-7 \times 2=
Get tutor helpright-arrow

Posted 10 months ago

Question
−1.2−2920= -1.2-\frac{29}{20}= \newlineEnter the answer as an exact decimal or simplified fraction.
Get tutor helpright-arrow

Posted 9 months ago

View all (31)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant