Sama : 5 to the power of eight times 5 to the power of four divided by (5 to the power of three) to the power of two

Question

Jimena: 2 to the power of nine times (2 to the power of negative two) to the power of negative three divided by 2

Who has more money in their account?
What is the difference in the amount of money that they have?

Answer 1

Let's calculate how much money Sama and Jimena have in their accounts by simplifying their expressions.

Sama's Calculation:

Sama's expression is: \[ \frac{5^8 \times 5^4}{(5^3)^2} \]

1. Simplify the numerator: \[ 5^8 \times 5^4 = 5^{8 + 4} = 5^{12} \]

2. Simplify the denominator: \[ (5^3)^2 = 5^{3 \times 2} = 5^6 \]

3. Now substitute back into the expression: \[ \frac{5^{12}}{5^6} = 5^{12 - 6} = 5^6 \]

Jimena's Calculation:

Jimena's expression is: \[ \frac{2^9 \times (2^{-2})^{-3}}{2} \]

1. Simplify the term \((2^{-2})^{-3}\): \[ (2^{-2})^{-3} = 2^{-2 \times -3} = 2^{6} \]

2. Now put this back into the expression: \[ 2^9 \times 2^6 = 2^{9 + 6} = 2^{15} \]

3. Now substitute back: \[ \frac{2^{15}}{2} = 2^{15 - 1} = 2^{14} \]

Now we compare \(5^6\) and \(2^{14}\).

1. Calculate \(5^6\): \[ 5^6 = 15625 \]

2. Calculate \(2^{14}\): \[ 2^{14} = 16384 \]

Conclusions:

• Sama has: 15625
• Jimena has: 16384

Who has more money?

Jimena has more money.

Difference in the amount of money:

To find the difference: \[ 16384 - 15625 = 759 \]

Thus, Jimena has more money than Sama, and the difference is 759.