Find the value of the expression: [tex]$0.5^{10} \div 0.5^7$[/tex]
Answer (1)
Use the quotient of powers property: a m ÷ a n = a m − n .
Apply the property to the given expression: 0. 5 10 ÷ 0. 5 7 = 0. 5 10 − 7 .
Simplify the exponent: 10 − 7 = 3 .
Evaluate 0. 5 3 = 0.125 .
The final answer is 0.125 .
Explanation
Understanding the Problem We are asked to find the value of the expression 0. 5 10 ÷ 0. 5 7 . This involves dividing two exponential terms that have the same base.
Applying the Quotient of Powers Property To solve this, we will use the quotient of powers property, which states that when dividing exponential terms with the same base, we subtract the exponents: a m ÷ a n = a m − n .
Simplifying the Exponent Applying this property to our expression, we get: 0. 5 10 ÷ 0. 5 7 = 0. 5 10 − 7 .
Calculating the Result Now, we simplify the exponent: 10 − 7 = 3 . So our expression becomes: 0. 5 3 .
Final Answer We can calculate 0. 5 3 as follows: 0. 5 3 = 0.5 × 0.5 × 0.5 = 0.125 . Alternatively, we can express 0.5 as a fraction: 0.5 = 2 1 . Therefore, 0. 5 3 = ( 2 1 ) 3 = 2 3 1 3 = 8 1 = 0.125 .
Examples
Understanding exponential division is useful in many areas, such as calculating compound interest or modeling population growth. For example, if a population doubles every year, the ratio of the population after 10 years to the population after 7 years would be 2 10 / 2 7 = 2 3 = 8 . This means the population after 10 years is 8 times the population after 7 years.
