Key: logx= log base x

1. Prove that:
(logx64 +logx4 -logx8)/(logx1024) =1/2

2.factorize by grouping

r^2s^2+3t^2+r^2t^2+3s^2

3.factorize the following expression

49p^2-1/25

4.simplify

i.(ax+bx+ay+by)/(cx+cy+dx+dy)

ii.(3/4+1/2) / (1/3+1/2)

5.two thirds of a number together with a quarter of the same number and a sixth of that number equal 52.calculate the number

Hints:

1.
Use basic laws of logarithm:
log(a)+log(b)=log(ab)
log(a)-log(b)=log(a/b)
log(a^b)=b(log(a))
and use the fact that all the numeric values are powers of 2:
2^6=64
2^2=4
2^3=8
2^10=1024

2.
I'll get you started:
r^2s^2+3t^2+r^2t^2+3s^2
=(r^2s^2+r^2t^2)+(3t^2+3s^2 )
take out common factors in each group enclosed in parentheses and continue factoring.

3.
Hint: expression is difference of two squares.
See for example:
http://www.regentsprep.org/regents/math/algebra/AV6/Lfactps.htm

4.
i. factorize numerator and denominator by grouping, and cancel common factors.
ii. do the sums of fractions and simplify.

5.
Let the unknown number be x.
Then interpret the question in terms of x. Solve for x.

1. To prove that (logx64 + logx4 - logx8) / (logx1024) = 1/2, we can simplify both the numerator and denominator using logarithmic properties.

Using the properties logb m + logb n = logb (m * n) and logb m - logb n = logb (m / n), we can simplify the numerator as follows:

(logx64 + logx4 - logx8) = logx(64 * 4 / 8) = logx32

Now, for the denominator, we have logx1024.

Using the property logb m^n = n * logb m, we can rewrite this as:

logx1024 = logx(2^10) = 10 * logx2

Now, we can substitute the simplified numerator and denominator back into the equation:

(logx64 + logx4 - logx8) / (logx1024) = (logx32) / (10 * logx2)

Now, we can further simplify by using the logarithmic property logb b = 1:

(logx32) / (10 * logx2) = logx32 / (logx2^10) = logx32 / (logx32) = 1/2

Therefore, (logx64 + logx4 - logx8) / (logx1024) = 1/2.

2. To factorize by grouping the expression r^2s^2 + 3t^2 + r^2t^2 + 3s^2, we group the terms in pairs:

(r^2s^2 + r^2t^2) + (3t^2 + 3s^2)

Now, we can factor out common terms from each pair:

r^2(s^2 + t^2) + 3(t^2 + s^2)

Finally, we can notice that (s^2 + t^2) is a sum of squares, which cannot be factorized further. Therefore, the factorized expression becomes:

r^2(s^2 + t^2) + 3(t^2 + s^2)

3. To factorize the expression 49p^2 - 1/25, we can rewrite 1/25 as (1/5)^2.

Now, we have 49p^2 - (1/5)^2. This is now in the form of a difference of squares.

Using the fact that a^2 - b^2 = (a + b)(a - b), we can apply this to our expression:

49p^2 - (1/5)^2 = (7p)^2 - (1/5)^2

Now, we have (7p + 1/5)(7p - 1/5) as the factorized expression.

4. i. To simplify the expression (ax + bx + ay + by) / (cx + cy + dx + dy), we can factor out common terms from the numerator and denominator:

(ax + bx + ay + by) / (cx + cy + dx + dy) = (a(x + y) + b(x + y)) / (c(x + y) + d(x + y))

Now, we have common factors of (x + y) in both the numerator and denominator, which can be canceled out:

(a + b) / (c + d)

Therefore, the simplified expression is (a + b) / (c + d).

ii. To simplify the expression (3/4 + 1/2) / (1/3 + 1/2), we can find a common denominator for both fractions:

(3/4 + 1/2) / (1/3 + 1/2) = (6/8 + 4/8) / (2/6 + 3/6)

Now, we can add the fractions:

(6/8 + 4/8) / (2/6 + 3/6) = 10/8 / 5/6

We can simplify the fraction on the left by dividing the numerator and denominator by their common factor of 2:

10/8 / 5/6 = (10/8) * (6/5) = 60/40

Finally, we can simplify the fraction by dividing the numerator and denominator by their common factor of 20:

60/40 = 3/2

Therefore, the simplified expression is 3/2.

5. Let's represent the number as "x".

According to the given information, we have the equation: (2/3)x + (1/4)x + (1/6)x = 52.

Now, let's combine the fractions by finding a common denominator:

[(2/3) + (1/4) + (1/6)]x = 52

The common denominator of 3, 4, and 6 is 12:

[(8/12) + (3/12) + (2/12)]x = 52

Now, let's add the fractions:

(13/12)x = 52

To solve for x, we need to isolate it. Multiply both sides of the equation by the reciprocal of (13/12), which is (12/13):

(13/12)x * (12/13) = 52 * (12/13)

x = 624/13

Therefore, the number is x = 624/13.

1. To prove the equation (logx64 + logx4 - logx8)/(logx1024) = 1/2, we can start by simplifying each logarithm.

a) logx64: This is the logarithm with base x that gives us 64. Therefore, x to what power equals 64? We can rewrite this as x^logx64 = 64, which means x^6 = 64.

b) logx4: Similarly, x^logx4 = 4. This means x^2 = 4.

c) logx8: x^logx8 = 8. This means x^3 = 8.

Now, let's substitute these simplified expressions back into the original equation:

(logx64 + logx4 - logx8)/(logx1024) = (x^6 + x^2 - x^3)/(x^logx1024)

Since logx1024 = 10 (because x^10 = 1024), the equation becomes:

(x^6 + x^2 - x^3)/(x^10) = 1/2

To simplify the equation further, we can multiply both sides by 2x^10:

2(x^6 + x^2 - x^3) = x^10

Expanding the left side:

2x^6 + 2x^2 - 2x^3 = x^10

Rearranging terms:

x^10 - 2x^6 - 2x^2 + 2x^3 = 0

Now, we need to prove that this equation is true for all values of x. However, this equation does not hold true for all values of x; it is not a valid equation. Therefore, the given statement cannot be proved.

2. To factorize the expression r^2s^2 + 3t^2 + r^2t^2 + 3s^2 using grouping, we can group like terms together:

(r^2s^2 + r^2t^2) + (3t^2 + 3s^2)

Now, we can factor out common terms from each group:

r^2(s^2 + t^2) + 3(t^2 + s^2)

Finally, we can factor out a common factor from both terms:

r^2(s^2 + t^2) + 3(t^2 + s^2) = (s^2 + t^2) (r^2 + 3)

3. To factorize the expression 49p^2 - 1/25, we can write it as the difference of two squares:

(7p)^2 - (1/5)^2

Using the identity a^2 - b^2 = (a + b)(a - b), we can factorize it further:

(7p + 1/5)(7p - 1/5)

4.
i. To simplify (ax + bx + ay + by)/(cx + cy + dx + dy), we can factor out common terms from the numerator and denominator:

(a(x + y) + b(x + y))/(c(x + y) + d(x + y))

Now, we can simplify further by dividing both the numerator and denominator by (x + y):

(a + b)/(c + d)

ii. To simplify (3/4 + 1/2) / (1/3 + 1/2), we can first add the fractions in the numerator and denominator:

(3/4 + 1/2) / (1/3 + 1/2) = (6/8 + 4/8) / (2/6 + 3/6)

Simplifying further:

(10/8) / (5/6) = (10/8) * (6/5) = 3/2

5. Let's represent the number as "x". The given equation states:

(2/3)x + (1/4)x + (1/6)x = 52

To solve for x, we can combine the fractions on the left side:

[(2/3) + (1/4) + (1/6)]x = 52

The least common denominator (LCD) of 3, 4, and 6 is 12. So, we can rewrite the equation with a common denominator:

[(8/12) + (3/12) + (2/12)]x = 52

Combining the fractions further:

(13/12)x = 52

To solve for x, we can multiply both sides of the equation by the reciprocal of (13/12), which is (12/13):

(12/13) * (13/12)x = (12/13) * 52

Simplifying:

x = 48

Therefore, the number is 48.