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

 24 is split into two parts such that 7 times the first plus 5 times the second is 146.

Let the first part be 

x, then the second is 

7x+5(24−x)=146

7x+120−5x=146

2x=26⇒x=13

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

1 square meter = 10.764 square feet (exact metric to imperial conversion).

This is used in construction, real estate, and area conversions internationally.

Explanation

1. Divide 3239 by 2923:

$$3239÷2923=1\text{(remainder = 3239 - 2923 = 316)}$$

2. Now, divide 2923 by 316:

$$2923÷316=9\text{(remainder = 2923 - 9 × 316 = 2923 - 2844 = 79)}$$

3. Now, divide 316 by 79:

$$316÷79=4\text{(remainder = 316 - 4 × 79 = 316 - 316 = 0)}$$

Since the remainder is 0, the H.C.F is 79.

Explanation

Simplify the square root:

√1.96/100

√1.96/ √100

√1.96 = 1.4

√100 =10

14/10 = 0.14

Explanation

We are given:

• H.C.F (Highest Common Factor) = 12

• Difference between the two numbers = 12

  Step-by-step method:

Let the two numbers be:
x and x + 12

Since H.C.F = 12, both numbers must be divisible by 12.

Let’s assume:

$$x=12a\text{and}x+12=12b$$

So,

$$x+12-x=12b-12a=12(b-a)=12\Rightarrow b-a=1$$

So:

$$a=k,b=k+1\Rightarrow \text{The numbers are }12k\text{ and }12(k+1)$$

Try k = 7:

$$12 × 7 = 84,quad 12 × 8 = 96 Rightarrow text{Numbers are } 84 text{ and } 96$$

• Difference: $96-84=12$ 

• H.C.F of 84 and 96 is 12 

Explanation

Given numbers:
0.1, 0.21, 0.31, 0.41, 0.51

Step 1: Add all the numbers

$$0.1 + 0.21 + 0.31 + 0.41 + 0.51 = 1.54$$

Step 2: Divide by the number of values

$$frac{1.54}{5} = 0.308$$

Explanation

Given:

• 16 machines finish the work in 4 days

• Now we want to know how many days will take 8 machines for the same work

Concept:

More machines → Less time
Fewer machines → More time
This is an inverse proportion relationship.

So,

$$text{Days} propto frac{1}{text{Machines}}$$ $$text{New Days} = 4 times frac{16}{8} = 4 times 2 = 8 text{ days}$$