Explanation

 We are given the equation:

30y=60

To find the value of y, divide both sides by 30:

𝑦=60/30=2

Explanation

If4^a=8^b, find the ratio $frac{a}{b}$

Write both sides with base 2:

$$4^a = (2^2)^a = 2^{2a} \ 8^b = (2^3)^b = 2^{3b} Rightarrow 2^{2a} = 2^{3b} Rightarrow 2a = 3b Rightarrow frac{a}{b} = frac{3}{2}$$

⇒2^2a=2^3b⇒2a=3b⇒a​/b=3/2​

Explanation

Given:

(1+x)(1+x2)(1+x4)(1+x8)(1−x),

$$$$(1+x)(1+x2)(1+x4)(1+x8)(1−x)

This is a well-known identity in algebra that builds up as a geometric progression product:

Identity:

(1−x)(1+x)(1+x^2)(1+x^4)(1+x^8)=1−x^16

$$$$(1−x)(1+x)(1+x2)(1+x4)(1+x8)=1−x16

So,

(1+x)(1+x^2)(1+x^4)(1+x^8)(1−x)=1−x^16

$$$$(1+x)(1+x2)(1+x4)(1+x8)(1−x)=1−x16

Explanation

If $a + 3b = 6$ and $ab=3$, then the value of $a^2 + 9b^2$ is:

We are given:

a+3b=6(1),ab=3(2)

$$$$a+3b=6(1),ab=3(2)

Now square equation (1):

$$(a + 3b)^2 = 6^2 = 36 Rightarrow a^2 + 9b^2 + 6ab = 36$$

Use $ab = 3$

$$a^2 + 9b^2 + 6(3) = 36 Rightarrow a^2 + 9b^2 + 18 = 36 Rightarrow a^2 + 9b^2 = 18$$

Explanation

Let the total number of rabbits = x

Let the total number of pigeons = y

total number of head = 20 

                x + y = 20        eq (1)

total number of feet = 48 

             4x  +2y  = 48

              2x + y   = 24    eq (2)

Subtracting equation (2) from eq (1)

   2x + y -x - y = 24 -20

        x = 4