Finding Unit Digit of Powers

To find the unit digit of ($489^{86} - 351^{63}$), we need to find the unit digit of each term separately.

Unit Digit of $489^{86}$

The unit digit depends only on the unit digit of the base, which is 9.

• The pattern of the unit digits of powers of 9 is: $9^1 \rightarrow 9$, $9^2 \rightarrow 1$, $9^3 \rightarrow 9$, $9^4 \rightarrow 1$, ...
• The pattern repeats every 2 powers. For an even exponent, the unit digit is 1.
• Since the exponent is 86 (an even number), the unit digit of $489^{86}$ is 1.

Unit Digit of $351^{63}$

The unit digit depends only on the unit digit of the base, which is 1.

• The unit digit of any positive integer power of 1 is always 1.
• Therefore, the unit digit of $351^{63}$ is 1.

Calculating the Difference Unit Digit

Now, we find the unit digit of the difference:

Unit digit of ($489^{86} - 351^{63}$) = Unit digit of (Unit digit of $489^{86}$ - Unit digit of $351^{63}$)

Unit digit of ($489^{86} - 351^{63}$) = Unit digit of ($1 - 1$)

Unit digit of ($489^{86} - 351^{63}$) = Unit digit of (0)

The unit digit is 0.