Math Snap

STEP 1

Assumptions1. The number given is $52.4_{8}$, which is in base eight (octal).
. We need to convert this number to base ten (decimal).

STEP 2

To convert a number from base eight to base ten, we need to multiply each digit of the number by $8^n$, where $n$ is the position of the digit from right to left, starting from0.
The formula for conversion is$$Decimal = d0 \times8^0 + d1 \times8^1 + d2 \times8^2 + ... + d_n \times8^n$$Where $d_n$ is the nth digit of the octal number.

STEP 3

First, we separate the integer part and the fractional part of the octal number.
The integer part is52 and the fractional part is.

STEP 4

We convert the integer part from octal to decimal.
Using the formula, we get$$Decimal_{integer} =2 \times8^0 + \times8^1$$

STEP 5

Calculate the decimal equivalent of the integer part.
$$Decimal_{integer} =2 \times1 +5 \times8 =2 +40 =42$$

STEP 6

Next, we convert the fractional part from octal to decimal.
The formula for converting the fractional part is similar to the integer part, but we use negative powers of8.
$$Decimal_{fraction} = d_{-1} \times8^{-1} + d_{-2} \times8^{-2} + ... + d_{-n} \times8^{-n}$$Where $d_{-n}$ is the nth digit of the fractional part of the octal number.

STEP 7

Apply the formula to the fractional part of the octal number.
$$Decimal_{fraction} =4 \times^{-1}$$

STEP 8

Calculate the decimal equivalent of the fractional part.
$$Decimal_{fraction} =4 \times0.125 =0.5$$

STEP 9

Now, we add the decimal equivalents of the integer part and the fractional part to get the decimal equivalent of the octal number.
$$Decimal = Decimal_{integer} + Decimal_{fraction}$$

STEP 10

Plug in the values for the decimal equivalents of the integer part and the fractional part.
$$Decimal =42 +0.5$$

Calculate the decimal equivalent of the octal number.
$$Decimal =42 +0.5 =42.5$$So, the number $52.4_{8}$ in base eight (octal) is equivalent to $42.5$ in base ten (decimal).