Simplify the expression:<br> (x^3y^n/z^m+1)^k

Let a_n be the integer obtained by writing all the integers from 1 to n from left to right. For example, a_3 = 123 and a_{11} =

Artemon [7]

Answer:

9

Step-by-step explanation:

The 79-digit number of interest can be formulated as a sum of shorter numbers whose remainders can be computed.

<h3>Expanded form</h3>

The expanded form of the number can be written as ...

a_44 = 01×10^78 +23×10^76 +45×10^74 +67×10^72 +89×10^70 +...

+10×10^68 +11×10^66 +... +43×10^2 +44×10^0

<h3>Powers of 10</h3>

Each number except the last is multiplied by a power of 10. Powers of 10 modulo 45 are ...

10 mod 45 = 10

100 mod 45 = 10

This lets us conclude that any positive power of 10 mod 45 is 10.

<h3>Parts of the sum</h3>

In short, all of the multiplication by powers of 10 can be collapsed to a single multiplication by 10. Hence, the mod 45 value of a_44 will be ...

a_44 mod 45 = (((01 +23 +45 +67 +89) +10 +11 +12 +... +43)×10 +44) mod 45

= (((01 +23 +45 +67 +89) mod 45 + sum(10 .. 43) mod 45)×10 +44) mod 45

= (((225 mod 45) +(901 mod 45))×10 +44) mod 45

= ((0 +1)×10 +44) mod 45

<h3>Final value</h3>

a_44 mod 45 = 54 mod 45 = 9

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<em>Additional comment</em>

The result can be confirmed by a suitable calculator.

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The sum of the 34 numbers from 10 to 43 is the product of their average value (10+43)/2 = 26.5 and their number, 34. (26.5×34) = 901.

3 0

2 years ago

Calculate the average rate of change from x = 0 to x = 3. Show work please

AURORKA [14]

$Rate=\frac{f(b)-f(a)}{b-a}=\frac{f(3)-f(0)}{3-0}=\frac{3-(-4)}{3}=\frac{7}{3}$

3 0

3 years ago

What is the definition of distributive property?

anyanavicka [17]

The distributive property is one of the most frequently used properties in math. In general, this term refers to the distributive property of multiplication which states that the. Definition: Thedistributive property lets you multiply a sum by multiplying each addend separately and then add the products.

7 0

4 years ago

Read 2 more answers

Rationalize the denominator of $\frac{5}{2+\sqrt{6}}$. The answer can be written as $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$

horrorfan [7]

Answer:

$A +B+C+D = 3$ is the correct answer.

Step-by-step explanation:

Given:

$\frac{5}{2+\sqrt{6}}$

To find:

$A+B+C+D = ?$ if given term is written as following:

$\frac{A\sqrt{B}+C}{D}$

<u>Solution:</u>

We can see that the resulting expression does not contain anything under $\sqrt$ (square root) so we need to rationalize the denominator to remove the square root from denominator.

The rule to rationalize is:

Any term having square root term in the denominator, multiply and divide with the expression by changing the sign of square root term of the denominator.

Applying this rule to rationalize the given expression:

$\dfrac{5}{2+\sqrt{6}} \times \dfrac{2-\sqrt6}{2-\sqrt6}\\\Rightarrow \dfrac{5 \times (2-\sqrt6)}{(2+\sqrt{6}) \times (2-\sqrt6)} \\\Rightarrow \dfrac{10-5\sqrt6}{2^2-(\sqrt6)^2}\ \ \ \ \ (\because \bold{(a+b)(a-b)=a^2-b^2})\\\Rightarrow \dfrac{10-5\sqrt6}{4-6}\\\Rightarrow \dfrac{10-5\sqrt6}{-2}\\\Rightarrow \dfrac{-5\sqrt6+10}{-2}\\\Rightarrow \dfrac{5\sqrt6-10}{2}$