Simplify the expression so there is only one power for each base.
[tex]5.6^{-5} \cdot 3.4^{-7} \cdot 5.6^3 \cdot 3.4^{-4}[/tex]
A. [tex]5.6^{-2} \cdot 3.4^{-11}[/tex]
B. [tex]5.6^8 \cdot 3.4^{11}[/tex]
C. [tex]5.6^{-8} \cdot 3.4^{-7}[/tex]
D. [tex]5.6^{-8} \cdot 3.4^{-3}[/tex]
E. [tex]5.6^{-5} \cdot 3.4^{-3}[/tex]
Answer (2)
Group terms with the same base: $(5.6^{-5}
cdot 5.6^3)
cdot (3.4^{-7}
cdot 3.4^{-4})$.
Apply the exponent rule $a^m
cdot a^n = a^{m+n}$ to the powers of 5.6: 5. 6 − 5 + 3 = 5. 6 − 2 .
Apply the exponent rule to the powers of 3.4: 3. 4 − 7 + ( − 4 ) = 3. 4 − 11 .
The simplified expression is $\boxed{5.6^{-2}
cdot 3.4^{-11}}$.
Explanation
Understanding the Problem We are asked to simplify the expression 5. 6 − 5 ⋅ 3. 4 − 7 ⋅ 5. 6 3 ⋅ 3. 4 − 4 so that there is only one power for each base. This involves using the properties of exponents to combine terms with the same base.
Using the Properties of Exponents The key property we'll use is that a m ⋅ a n = a m + n . This means when multiplying terms with the same base, we add their exponents.
Grouping Like Terms First, let's group the terms with the same base: ( 5. 6 − 5 ⋅ 5. 6 3 ) ⋅ ( 3. 4 − 7 ⋅ 3. 4 − 4 )
Simplifying Powers of 5.6 Now, we apply the exponent rule to each group: For the base 5.6: 5. 6 − 5 ⋅ 5. 6 3 = 5. 6 − 5 + 3 = 5. 6 − 2
Simplifying Powers of 3.4 For the base 3.4: 3. 4 − 7 ⋅ 3. 4 − 4 = 3. 4 − 7 + ( − 4 ) = 3. 4 − 11
Final Simplified Expression So, the simplified expression is: 5. 6 − 2 ⋅ 3. 4 − 11
Identifying the Correct Option Comparing our simplified expression 5. 6 − 2 ⋅ 3. 4 − 11 with the given options, we see that it matches option A.
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. For example, if you invest money that grows at a certain percentage each year, you can use exponents to calculate the future value of your investment. Similarly, in science, exponential decay is used to model the decrease in the amount of a radioactive substance over time. The ability to manipulate exponents allows for easier calculations and a better understanding of these phenomena.
The expression 5. 6 − 5 ⋅ 3. 4 − 7 ⋅ 5. 6 3 ⋅ 3. 4 − 4 simplifies to 5. 6 − 2 ⋅ 3. 4 − 11 , which corresponds to option A. This is achieved by combining like terms and using the exponent rules effectively. The final answer is option A: 5. 6 − 2 ⋅ 3. 4 − 11 .
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