Simplify the expression shown below:
[tex]$\left(a^2 b^8 c^5\right)^2\left(a^5 c^2\right)^3\left(a^{-2} b c^{-1}\right)^3$[/tex]
A. [tex]$a^{13} b^{19} c^{13}$[/tex]
B. [tex]$a^{21} b^{24} c^{10}$[/tex]
C. [tex]$6\left(a^8 b^{12} c^4\right)$[/tex]
D. [tex]$a^{21} b^{29} c^{14}$[/tex]
Answer (2)
Apply the power of a product rule to each term.
Multiply the simplified terms together by adding the exponents of like bases.
Simplify the exponents.
The simplified expression is a 13 b 19 c 13 .
Explanation
Understanding the Problem We are asked to simplify the expression ( a 2 b 8 c 5 ) 2 ( a 5 c 2 ) 3 ( a − 2 b c − 1 ) 3 . To do this, we will use the rules of exponents.
Applying the Power of a Product Rule First, we apply the power of a product rule to each term: ( x y ) n = x n y n . This gives us: \begin{align*}\left(a^2 b^8 c^5\right)^2 &= a^{2\times2} b^{8\times2} c^{5\times2} = a^4 b^{16} c^{10} \ \left(a^5 c^2\right)^3 &= a^{5\times3} c^{2\times3} = a^{15} c^6 \ \left(a^{-2} b c^{-1}\right)^3 &= a^{-2\times3} b^{1\times3} c^{-1\times3} = a^{-6} b^3 c^{-3}\end{align*}
Multiplying the Simplified Terms Next, we multiply the simplified terms together. When multiplying terms with the same base, we add their exponents. So we have: \begin{align*}a^4 b^{16} c^{10} \times a^{15} c^6 \times a^{-6} b^3 c^{-3} &= a^{4+15-6} b^{16+3} c^{10+6-3} \ &= a^{13} b^{19} c^{13}\end{align*}
Final Answer Therefore, the simplified expression is a 13 b 19 c 13 .
Examples
Understanding how to simplify expressions with exponents is useful in many fields, such as physics and computer science. For example, in physics, you might use these rules when dealing with scientific notation or calculating the energy of a photon. In computer science, you might use these rules when analyzing the complexity of algorithms or working with data compression techniques. Simplifying expressions helps make complex calculations more manageable and understandable.
We simplified the expression ( a 2 b 8 c 5 ) 2 ( a 5 c 2 ) 3 ( a − 2 b c − 1 ) 3 to obtain a 13 b 19 c 13 , which corresponds to option A. This involved applying the power of a product rule and combining like bases by adding their exponents. The final answer is also provided in the options given.
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