What is the missing number in the simplified exponential expression 8 / 8 ^ 4 = 8 ^ 15
The missing number in the simplified exponential expression is 4.
To find the missing number in the simplified exponential expression, we need to solve for the exponent.
Let's start by simplifying the left side of the equation, 8 / 8 ^ 4:
8 / 8 ^ 4 = 8 / (8 * 8 * 8 * 8)
Since 8 ^ 4 means 8 raised to the power of 4, we can rewrite the denominator as:
8 / (8 * 8 * 8 * 8) = 8 / (8 ^ 4)
Now, let's simplify the right side of the equation, which is 8 ^ 15:
8 ^ 15 means 8 raised to the power of 15.
Since the base is the same (8), we can equate the exponents:
8 ^ 15 = 8 ^ 4
Therefore, the missing number is 4.
To find the missing number in the simplified exponential expression, we need to apply the properties of exponents.
The expression is: 8 / 8 ^ 4 = 8 ^ 15
Let's break it down step by step:
1. Start by simplifying the expression on the left side of the equation. We have a base of 8 with a negative exponent of 4:
8 ^ (-4)
2. Applying the rule for negative exponents, we can rewrite 8 ^ (-4) as 1 / 8 ^ 4:
1 / 8 ^ 4
3. Now that both sides of the equation have the same denominator, we can equate the numerators:
1 = 8 ^ 15
4. To isolate the base on the right side of the equation, we need to take the logarithm of both sides. Let's assume we are taking the natural logarithm (ln) for this explanation:
ln(1) = ln(8 ^ 15)
5. According to the logarithmic property, we can bring the exponent down as a coefficient:
ln(1) = 15 * ln(8)
6. The natural logarithm of 1 is equal to 0:
0 = 15 * ln(8)
7. Finally, divide both sides of the equation by 15 to solve for ln(8):
ln(8) = 0 / 15
ln(8) = 0
Therefore, the missing number is 0.