Let's go through both the questions from the image you provided.

Problem 19:

Find the smallest number by which 256 must be multiplied to obtain a perfect cube.

To solve this, we start by finding the prime factorization of 256: $256 = 2^8$ To form a perfect cube, the powers of all prime factors must be multiples of 3. Since $2^8$ has a power of 8, we need to multiply by $2^1$ to make the exponent a multiple of 3: $8 + 1 = 9 \quad \text{(which is divisible by 3)}$ Thus, we need to multiply 256 by $2^1 = 2$.

Answer: The smallest number to multiply by is 2.

Problem 20:

In the parallelogram RUNS, find the values of x and y (lengths in cm).

The parallelogram has a diagonal marked with values involving $x$ and $y$:

• $RN = x + y$
• $UN = y + 4$

It is given that diagonals of a parallelogram bisect each other, meaning the lengths of $RN$ and $UN$ will be equal: $x + y = y + 4$

Now, subtract $y$ from both sides: $x = 4$

Once we have $x = 4$, substitute it into the equation $RN = x + y$: $4 + y = y + 4$

Thus, the value of $x$ is 4 and you can find $y$ by solving the equations or determining it's a unique solution from symmetry, but with all lengths in cm.

Would you like more details on any of these problems? Here are some related questions for further practice:

1. What is the prime factorization of 512?
2. Can you explain why diagonals of a parallelogram bisect each other?
3. What are the properties of a perfect square compared to a perfect cube?
4. How would you find the cube root of 4096?
5. What happens when the diagonals of a parallelogram are not equal?

Tip: When solving geometry problems, always consider the properties of the shape, such as symmetry or equal sides, as they often provide clues to the solution.