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Aloiza [94]
3 years ago
14

Prove that 17^16+17^14 is divisible by 5

Mathematics
1 answer:
fredd [130]3 years ago
6 0

We'll use the exponent rule that a^(b+c) = a^b*a^c

to break up 17^16 into 17^(14+2) = 17^14*17^2

That allows us to factor like so

17^16+17^14

17^14*17^2+17^14*1

17^14(17^2 + 1)

Computing 17^14 is a lot more difficult compared to computing 17^2 + 1, assuming you only could use paper and pencil.

Note that 17^2 = 17*17 = 289, so 17^2 + 1 = 289+1 = 290

Because 290 ends in a 0, this means that 5 is a factor

290 = 5*58

This makes 5 to be a factor of the original expression.

----------------

So overall,

17^16+17^14 = 17^14(17^2 + 1) = 17^14*290 = 17^14*5*58 = 5*17^14*58

The stuff "17^14*58" doesn't matter. All we care about is that factor of 5.

Since 5 is a factor, this means that 17^16+17^14 is a multiple of 5, or 17^16+17^14 is divisible by 5.

Phrased another way, 17^16+17^14 all divided over 5 leads to some integer. In this case, the integer would be the result of 17^14*58.

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In isosceles right triangle ABC, point is on hypotenuse \overline{BC} such that \overline{AD} is an altitude of \triangle ABC an
Dmitriy789 [7]

Answer:

Area of triangle is 25.

Step-by-step explanation:

We have been given an isosceles right triangle

Isosceles triangle is the triangle having two sides equal.

Figure is shown in attachment

By Pythagoras theorem

BC^2=AC^2+AB^2

AD is altitude which divides the triangle into two parts

DC=5 implies BC =10 since D equally divides BC

Let AC=a implies AB=a being Isosceles

On substituting the values in the Pythagoras theorem:

10^2=a^2+a^2

100=2a^2

\Rightarrow a^2=50

\Rightarrow a=\pm5\sqrt{2}

WE can find area of right triangle by considering height AB and AD

Area of triangle ABC is:

\frac{1}{2}\cdot BC\cdot AD     (1)

\Rightarrow \frac{1}{2}\cdot 10\cdot AD

And other method of area of triangle is:

\frac{1}{2}\cdot AB\cdot BC       (2)

Equating (1) and (2) we get:

\frac{1}{2}\cdot 10\cdot AD=\frac{1}{2}\cdot a\cdot a

\Rightarrow AD=\frac{a^2}{10}

\Rightarrow AD=\frac{50}{10}=5

Using area of triangle is: \frac{1}{2}\cdot BC\cdot AD

Now, the area of triangle ABC=\frac{1}{2}\cdot 5\cdot 10

\Rightarrow 25



4 0
3 years ago
What is the length side of a triangle that has vertices at (-5, -1), (-5, 5), and (3, -1)?
Rasek [7]

The side lengths of triangle are 6 units, 8 units and 10 units.

<u>SOLUTION: </u>

Given that, we have to find what is the length side of a triangle that has vertices at (-5, -1), (-5, 5), and (3, -1)  

We know that, distance between two points P\left(x_{1}, y_{1}\right) \text { and } Q\left(x_{2}, y_{2}\right) is given by  

P Q=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}

Now,  

\begin{array}{l}{\text { Distance between }(-5,-1) \text { and }(-5,5)=\sqrt{(-5-(-5))^{2}+(5-(-1))^{2}}} \\\\ {\qquad \begin{array}{l}{=\sqrt{(-5-(-5))^{2}+(5-(-1))^{2}}} \\\\ {=\sqrt{(0)^{2}+(5+1)^{2}}=\sqrt{(6)^{2}}=6} \\\\ {=\sqrt{(-5)^{2}+(5+1)^{2}+(5-(-1))^{2}}} \\\\ {=\sqrt{(-8)^{2}+(5+1)^{2}}=\sqrt{64+36}=\sqrt{100}=10} \\\\ {=\sqrt{(3-1)^{2}+(-1-(-1))^{2}}} \\\\ {=\sqrt{(5+3)^{2}+(0)^{2}}=\sqrt{(8)^{2}}=8}\end{array}}\end{array}

8 0
3 years ago
Simplify the expression below. 2^3 x 2^2
Zigmanuir [339]

Answer:

32 or~2^5

Step-by-step explanation:

→ (2^3)(2^2)

=2^3*2^2

=(2*2*2)*(2*2)

=2*2*2*2*2

=32

<u><em>You can also put this in another way:</em></u>

<em />2^5=2^3*2^2=2^3^+^2=2^5<em />

7 0
3 years ago
Read 2 more answers
Which figure has the longer side and by how much , a square with an area of 81ft ^2 or a square with a perimeter of 84 ft
stellarik [79]
The square that has a perimeter of 84 ft has the longer side.
4 0
3 years ago
Will mark BRAINLIEST if you help
nlexa [21]
The first box is -7 and the other one is just 7
7 0
2 years ago
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