Simplify $4.2^{-3} \cdot 5.1^2 \cdot 4.2^4 \cdot 5.1^3$.
Answer (2)
Group terms with the same base and apply the exponent rule a m c d o t a n = a m + n .
Simplify the exponents: 4. 2 − 3 + 4 c d o t 5. 1 2 + 3 = 4. 2 1 c d o t 5. 1 5 .
Calculate 5. 1 5 = 3450.25251 .
Multiply the results: 4.2 c d o t 3450.25251 = 14491.060542 . The simplified expression is 14491.060542 .
Explanation
Understanding the Problem We are given the expression 4. 2 − 3 c d o t 5. 1 2 c d o t 4. 2 4 c d o t 5. 1 3 to simplify. This expression involves products of powers with the same base, which we can combine using the property a m c d o t a n = a m + n .
Combining Terms with the Same Base First, we group the terms with the same base: ( 4. 2 − 3 c d o t 4. 2 4 ) c d o t ( 5. 1 2 c d o t 5. 1 3 ) Now, we apply the property a m c d o t a n = a m + n to each group: 4. 2 − 3 + 4 c d o t 5. 1 2 + 3 4. 2 1 c d o t 5. 1 5
Calculating the Exponents Now we need to calculate 5. 1 5 . The result of this operation is: 5. 1 5 = 3450.25251 So the expression becomes: 4.2 c d o t 3450.25251
Final Calculation Finally, we multiply 4.2 by 3450.25251 :
4.2 c d o t 3450.25251 = 14491.060542 Therefore, the simplified expression is approximately 14491.060542 .
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating compound interest or dealing with exponential growth or decay in science. For example, if you invest money with compound interest, you might need to simplify expressions like ( 1 + r ) n , where r is the interest rate and n is the number of compounding periods. Similarly, in physics, you might encounter exponential decay when studying radioactive materials, where you need to simplify expressions involving exponents to determine the amount of material remaining after a certain time.
We simplified the expression 4. 2 − 3 ⋅ 5. 1 2 ⋅ 4. 2 4 ⋅ 5. 1 3 by combining like terms using exponent rules, resulting in 4.2 ⋅ 5. 1 5 , which evaluates to approximately 14491.060542 .
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