If 251m = 100two. Find the value of m

For $251_m$:

For $100_{\text{two}}$:

We can use factoring.
Find two numbers that multiply to $2 \times (-3) = -6$ and add to 5. These numbers are 6 and $-1$.

This gives two possible values for $m$:
$2m - 1 = 0 \implies 2m = 1 \implies m = \frac{1}{2}$
$m + 3 = 0 \implies m = -3$

A number base must be a positive integer greater than the largest digit used in the number. In $251_m$, the largest digit is 5, so $m$ must be greater than 5.
Neither $\frac{1}{2}$ nor $-3$ satisfy this condition. However, in some contexts, $m$ is simply a variable in an equation, and the base interpretation is secondary. Given the context of "Find the value of m", we provide the mathematical solutions. If this were strictly a number base problem, there would be no valid integer solution for $m > 5$. Assuming it's a general algebraic problem:

1b. Simplify the following; $\frac{0.00047 \times 0.0093}{0.00000065}$. Correct to 3 d.p.

To simplify, multiply numerator and denominator by $10^8$ to remove decimals:

The fourth decimal place is 6, so we round up the third decimal place.

2a. $36^{1/2} \times 64^{1/2} \times 5^0$

$36^{1/2} = \sqrt{36} = 6$
$64^{1/2} = \sqrt{64} = 8$
$5^0 = 1$ (Any non-zero number raised to the power of 0 is 1)

2b. $4^{-1/2} \times 16^{3/4} \times 32^{-2/5}$

$4^{-1/2} = \frac{1}{4^{1/2}} = \frac{1}{\sqrt{4}} = \frac{1}{2}$
$16^{3/4} = (\sqrt[4]{16})^3 = (2)^3 = 8$
$32^{-2/5} = \frac{1}{32^{2/5}} = \frac{1}{(\sqrt[5]{32})^2} = \frac{1}{(2)^2} = \frac{1}{4}$

3a. $5(\sqrt{1.6} \times \sqrt{0.4})$

3b. $\frac{2\sqrt{6} - \sqrt{3}}{3}$

The terms in the numerator, $2\sqrt{6}$ and $\sqrt{3}$, are not like terms, so they cannot be combined.
The denominator is 3. We can write this as two separate fractions if desired, but it doesn't simplify further.

Since no further simplification is possible, the expression remains as is.

4. With the aid of diagram explain the following;

i) Set (ii) Singleton Set (iii) Venn diagram

i) Set
A set is a well-defined collection of distinct objects. These objects are called elements or members of the set. Sets are typically denoted by capital letters, and their elements are listed within curly braces.
Example: The set of even numbers less than 10 is $A = \{2, 4, 6, 8\}$.
Diagram:

  A
+---+
| 2 |
| 4 |
| 6 |
| 8 |
+---+

ii) Singleton Set
A singleton set (or unit set) is a set containing exactly one element.
Example: The set of even prime numbers is $B = \{2\}$.
Diagram:

  B
+---+
| 2 |
+---+

iii) Venn diagram
A Venn diagram is a visual representation used to show the relationships between different sets. It uses overlapping circles or other shapes to illustrate the logical relationships between sets, with the universal set often represented by a rectangle.
Example: Let $U = \{1, 2, 3, 4, 5, 6\}$, $P = \{1, 2, 3\}$, and $Q = \{3, 4, 5\}$.
Diagram:

+---------------------------------+
| U                               |
|   +-------+     +-------+     |
|   |   P   |     |   Q   |     |
|   | 1   2 |-----| 4   5 |     |
|   |       |  3  |       |     |
|   +-------+     +-------+     |
|                                 |
+---------------------------------+

In this diagram, the rectangle represents the universal set $U$. Circle $P$ contains elements $1, 2, 3$. Circle $Q$ contains elements $3, 4, 5$. The overlap shows element 3, which is common to both sets $P$ and $Q$. Element 6 is in $U$ but not in $P$ or $Q$.