Questions: Which choice is equivalent to the expression below? 4^8.869 A. 4^(8+8 / 10+6 / 10+9 / 1000) B. 4^8 * 4^(8 / 10) * 4^(6 / 100) * 4^(9 / 1000) C. 4^8 * 4^(86 / 10) * 4^(9 / 100) D. 4^8+4^(8 / 10)+4^(6 / 100)

Solution Steps

To solve this problem, we need to express the given exponent \(8.869\) in a form that matches one of the provided choices. The key is to break down the decimal into a sum of fractions that can be expressed as powers of 4. We will convert the decimal \(8.869\) into a sum of fractions and then use the properties of exponents to match it with one of the given options.

The given expression is \(4^{8.869}\). We need to express the decimal part \(0.869\) as a sum of fractions. The integer part is \(8\), and the decimal part is approximately \(0.869\).

The decimal \(0.869\) can be broken down as follows:

• Tenths place: \(8\), which is \(\frac{8}{10}\)
• Hundredths place: \(6\), which is \(\frac{6}{100}\)
• Thousandths place: \(9\), which is \(\frac{9}{1000}\)

The exponent \(8.869\) can be expressed as: \[ 8 + \frac{8}{10} + \frac{6}{100} + \frac{9}{1000} \]

Using the properties of exponents, we can rewrite the expression: \[ 4^{8.869} = 4^{8} \cdot 4^{\frac{8}{10}} \cdot 4^{\frac{6}{100}} \cdot 4^{\frac{9}{1000}} \]

Final Answer