6. $p^{x} \times p^{y} = p^{z}$ (a) Find the value of $x$ - Edexcel - GCSE Maths - Question 6 - 2017 - Paper 2

In the expression $(7)^{y} = (7)^{h}$, since the bases are the same, we can equate the exponents. Thus, we can write:

$y = h$

Therefore, the value of $y$ is simply equal to $h$, which is determined by the context of the problem or given information.

To express $100^{x} \times 1000^{b}$ in terms of powers of 10, we rewrite $100$ and $1000$ as powers of 10:

$100 = 10^{2} \quad \text{and} \quad 1000 = 10^{3}$

Then we substitute these values into the original expression:

$100^{x} = (10^{2})^{x} = 10^{2x}$ $1000^{b} = (10^{3})^{b} = 10^{3b}$

Combining these, we have:

$100^{x} \times 1000^{b} = 10^{2x} \times 10^{3b}$

Using the property of exponents (adding the exponents when multiplying), this simplifies to:

$10^{2x + 3b}$

To express this in the form $10^{n}$, we can set:

$n = 2x + 3b$

Assuming $x = a$, the equation becomes: $n = 2a + 3b$

This verifies the relationship as required.