\(\left(-2 k^3 \div q^4\right)^5 \quad\left(\left(\frac{1}{3}\) \left(c^5 \div f^6\right)\right)^2\)
Answer (2)
Simplify the first term ( − 2 k 3 ÷ q 4 ) 5 to q 20 − 32 k 15 .
Simplify the second term ( ( 3 1 ) ( c 5 ÷ f 6 ) ) 2 to 9 f 12 c 10 .
Multiply the simplified terms: q 20 − 32 k 15 ⋅ 9 f 12 c 10 .
The final simplified expression is 9 q 20 f 12 − 32 k 15 c 10 .
Explanation
Initial Analysis We are given the expression ( − 2 k 3 ÷ q 4 ) 5 ( ( 3 1 ) ( c 5 ÷ f 6 ) ) 2 and our goal is to simplify it. We will use exponent rules to simplify each term separately and then combine them.
Simplifying the First Term First, let's simplify the term ( − 2 k 3 ÷ q 4 ) 5 . This can be rewritten as ( q 4 − 2 k 3 ) 5 . Using the power of a quotient rule, we get ( q 4 ) 5 ( − 2 ) 5 ( k 3 ) 5 Now, we simplify the powers: ( − 2 ) 5 = − 32 , ( k 3 ) 5 = k 3 × 5 = k 15 , and ( q 4 ) 5 = q 4 × 5 = q 20 . So the first term simplifies to q 20 − 32 k 15 .
Simplifying the Second Term Next, let's simplify the second term ( ( 3 1 ) ( c 5 ÷ f 6 ) ) 2 . This can be rewritten as ( 3 1 ⋅ f 6 c 5 ) 2 = ( 3 f 6 c 5 ) 2 . Using the power of a quotient rule, we get ( 3 f 6 ) 2 ( c 5 ) 2 Now, we simplify the powers: ( c 5 ) 2 = c 5 × 2 = c 10 and ( 3 f 6 ) 2 = 3 2 ( f 6 ) 2 = 9 f 6 × 2 = 9 f 12 . So the second term simplifies to 9 f 12 c 10 .
Multiplying the Terms Now, we multiply the simplified first and second terms: q 20 − 32 k 15 ⋅ 9 f 12 c 10 = 9 q 20 f 12 − 32 k 15 c 10 .
Final Result Therefore, the simplified expression is 9 q 20 f 12 − 32 k 15 c 10 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the energy of a photon, E = h f , where h is Planck's constant and f is the frequency, if you have multiple photons or complex frequencies, you'll need to simplify expressions with exponents to find the total energy. Similarly, in electrical engineering, when dealing with impedances in AC circuits, you often encounter complex expressions involving powers and quotients that need simplification.
The expression ( − 2 k 3 ÷ q 4 ) 5 ( ( 3 1 ) ( c 5 ÷ f 6 ) ) 2 simplifies to 9 q 20 f 12 − 32 k 15 c 10 . This is achieved by applying the power of a quotient rule and simplifying each term separately before combining them. The final result is obtained through multiplication of the simplified terms.
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