Question 5:
Écrivez chacune des expressions suivantes sous la forme de la plus petite base possible affectée d'un exposant positif.
[tex]$5^{100} \times 25^{200}$[/tex]
Answer (1)
Rewrite 25 as a power of 5: 25 = 5 2 .
Substitute into the expression: 5 100 \t × ( 5 2 ) 200 .
Simplify using the power of a power rule: ( 5 2 ) 200 = 5 400 .
Simplify using the product of powers rule: 5 100 \t × 5 400 = 5 500 . The final answer is 5 500 .
Explanation
Understanding the Problem We are asked to write the expression 5 100 × 2 5 200 in the form of the smallest possible base with a positive exponent. This means we want to express the given expression as b e , where b is the smallest possible base and e is a positive exponent.
Rewriting the Expression First, we need to rewrite 25 as a power of 5 . Since 25 = 5 2 , we can substitute this into the expression: 5 100 × 2 5 200 = 5 100 × ( 5 2 ) 200 .
Applying the Power of a Power Rule Next, we simplify the expression using the power of a power rule, which states that ( a m ) n = a m × n . Applying this rule, we get ( 5 2 ) 200 = 5 2 × 200 = 5 400 .
Substituting Back into the Expression Now, we substitute this back into the original expression: 5 100 × 5 400 .
Applying the Product of Powers Rule We simplify the expression using the product of powers rule, which states that a m × a n = a m + n . Applying this rule, we get 5 100 × 5 400 = 5 100 + 400 = 5 500 .
Final Answer The expression is now in the form of the smallest possible base (5) with a positive exponent (500). Therefore, the final answer is 5 500 .
Examples
Understanding exponents is crucial in many fields, such as computer science and finance. For example, when calculating compound interest, the formula involves raising the interest rate plus one to the power of the number of compounding periods. Similarly, in computer science, understanding powers of 2 is essential for working with binary data and memory allocation. This problem demonstrates how to simplify expressions with exponents, a fundamental skill in these areas.
