\frac{-(-7)^0 \times 7^9}{(-7)^2(-7)^3(-7)}
Answer (1)
Simplify ( − 7 ) 0 to 1 .
Simplify the denominator ( − 7 ) 2 ( − 7 ) 3 ( − 7 ) to ( − 7 ) 6 , which equals 7 6 .
Rewrite the expression as 7 6 − 7 9 .
Apply the quotient rule to get − 7 9 − 6 = − 7 3 = − 343 .
Explanation
Understanding the Problem We are asked to simplify the expression ( − 7 ) 2 ( − 7 ) 3 ( − 7 ) − ( − 7 ) 0 × 7 9 . Let's break it down step by step, using exponent rules to make it easier to manage.
Simplifying the Zero Exponent First, we simplify ( − 7 ) 0 . Any non-zero number raised to the power of 0 is 1. Therefore, ( − 7 ) 0 = 1 .
Simplifying the Denominator Next, we simplify the denominator. We have ( − 7 ) 2 ( − 7 ) 3 ( − 7 ) . Using the product of powers rule, which states that a m × a n = a m + n , we can combine these terms: ( − 7 ) 2 ( − 7 ) 3 ( − 7 ) = ( − 7 ) 2 + 3 + 1 = ( − 7 ) 6 .
Rewriting the Expression Now, we rewrite the entire expression: ( − 7 ) 6 − 1 × 7 9 . Since ( − 7 ) 6 is the same as 7 6 because a negative number raised to an even power becomes positive, we have 7 6 − 7 9 .
Applying the Quotient Rule Using the quotient rule of exponents, which states that a n a m = a m − n , we simplify the expression further: 7 6 − 7 9 = − 7 9 − 6 = − 7 3 .
Calculating the Final Value Finally, we calculate − 7 3 . Since 7 3 = 7 × 7 × 7 = 343 , the final simplified expression is − 343 .
Examples
Understanding and simplifying exponential expressions is crucial in various fields, such as calculating compound interest, where the exponent represents the number of compounding periods. For instance, if you invest money in a bank account with compound interest, you need to understand how exponents work to calculate the future value of your investment. Similarly, in physics, exponential decay is used to model the decrease in the amount of a radioactive substance over time. Simplifying expressions with exponents helps in making accurate predictions and managing resources effectively.
