The problem asks us to compare two numbers expressed in expanded form using powers of 10 and insert the correct symbol (=, <, or >) between them. (i) Compare $8 \times 10^4 + 4 \times 10^3 + 7 \times 10^2 + 2 \times 10^1 + 9 \times 10^0$ and $9 \times 10^3 + 8 \times 10^1$. (ii) Compare $9 \times 10^2 + 2 \times 10^1 + 3 \times 10^0$ and $8 \times 10^2 + 9 \times 10^1 + 7 \times 10^0$.

1. Problem Description

The problem asks us to compare two numbers expressed in expanded form using powers of 10 and insert the correct symbol (=, <, or >) between them.

(i) Compare $8 \times 10^4 + 4 \times 10^3 + 7 \times 10^2 + 2 \times 10^1 + 9 \times 10^0$ and $9 \times 10^3 + 8 \times 10^1$.

(ii) Compare $9 \times 10^2 + 2 \times 10^1 + 3 \times 10^0$ and $8 \times 10^2 + 9 \times 10^1 + 7 \times 10^0$.

2. Solution Steps

(i)

First, evaluate the left-hand side (LHS):

$8 \times 10^4 + 4 \times 10^3 + 7 \times 10^2 + 2 \times 10^1 + 9 \times 10^0 = 8 \times 10000 + 4 \times 1000 + 7 \times 100 + 2 \times 10 + 9 \times 1 = 80000 + 4000 + 700 + 20 + 9 = 84729$

Then, evaluate the right-hand side (RHS):

$9 \times 10^3 + 8 \times 10^1 = 9 \times 1000 + 8 \times 10 = 9000 + 80 = 9080$

Since $84729 > 9080$, the symbol is $>$.

(ii)

First, evaluate the LHS:

$9 \times 10^2 + 2 \times 10^1 + 3 \times 10^0 = 9 \times 100 + 2 \times 10 + 3 \times 1 = 900 + 20 + 3 = 923$

Then, evaluate the RHS:

$8 \times 10^2 + 9 \times 10^1 + 7 \times 10^0 = 8 \times 100 + 9 \times 10 + 7 \times 1 = 800 + 90 + 7 = 897$

Since $923 > 897$, the symbol is $>$.

3. Final Answer

(i) $8 \times 10^4 + 4 \times 10^3 + 7 \times 10^2 + 2 \times 10^1 + 9 \times 10^0 > 9 \times 10^3 + 8 \times 10^1$

(ii) $9 \times 10^2 + 2 \times 10^1 + 3 \times 10^0 > 8 \times 10^2 + 9 \times 10^1 + 7 \times 10^0$