Find the value of the expression: [tex]$10^{15} \div 10^{12}$[/tex]
Answer (1)
Use the quotient of powers property: a n a m = a m − n .
Apply the property to the expression: 1 0 12 1 0 15 = 1 0 15 − 12 .
Simplify the exponent: 15 − 12 = 3 , so we have 1 0 3 .
Calculate the final value: 1 0 3 = 1000 , so the answer is 1000 .
Explanation
Understanding the Problem We are asked to find the value of the expression 1 0 15 ÷ 1 0 12 . This involves dividing two powers of 10.
Applying the Quotient of Powers Property To solve this, we will use the quotient of powers property, which states that when dividing powers with the same base, we subtract the exponents:
a n a m = a m − n
In our case, a = 10 , m = 15 , and n = 12 .
Simplifying the Exponent Applying the property to the given expression, we have:
1 0 12 1 0 15 = 1 0 15 − 12
Now, we simplify the exponent: 15 − 12 = 3 .
Calculating the Final Value Therefore, the expression simplifies to 1 0 3 . This means 10 multiplied by itself three times:
1 0 3 = 10 × 10 × 10 = 1000
Final Answer Thus, the value of the expression 1 0 15 ÷ 1 0 12 is 1000 .
Examples
Understanding exponents is crucial in many real-world scenarios. For example, in computer science, data storage is often measured in bytes, kilobytes, megabytes, gigabytes, and terabytes, which are all powers of 2 (similar to powers of 10). Calculating the difference in storage capacity between, say, a gigabyte and a terabyte, involves understanding and manipulating exponents. This also applies to calculating exponential growth in populations or decay in radioactive materials, where understanding exponential relationships is essential.
