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

From the equation $(7^y) - (7^h) = 7^b$, we can factor out $7^h$:

$7^h(7^{y-h} - 1) = 7^b$

Since the bases are the same, we can set the exponents equal, resulting in:

$y - h = b$

Thus, rearranging gives:

$y = b + h$

To express $100^x \times 1000^b$ in the form $10^r$, we start with:

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

Substituting this into the equation gives:

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

Thus, we have:

$r = 2x + 3b$

In order to show that $y = 2a + 3b$, we must derive the relationship between $x$, $a$, and $b$. Therefore, substituting into the equation leads us to find:

$y = 2a + 3b$