Which factor should be considered when choosing a mental health professional?
A. ment philosophy
B. marital status
Answer (1)
Rewrite the expression using exponents.
Express all terms as powers of their prime factors.
Simplify the exponents.
Combine the powers of 2 and find a common denominator.
The simplified expression is 2 105 241 ⋅ 3 5 1 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 3 4 5 6 7 8 . This involves rewriting the radicals as exponents and combining like terms.
Rewriting with Exponents First, rewrite the expression using exponents: 2 ⋅ 4 3 1 ⋅ 6 5 1 ⋅ 8 7 1
Prime Factorization Next, express all terms as powers of their prime factors: 2 ⋅ ( 2 2 ) 3 1 ⋅ ( 2 ⋅ 3 ) 5 1 ⋅ ( 2 3 ) 7 1
Simplifying Exponents Now, simplify the exponents: 2 ⋅ 2 3 2 ⋅ 2 5 1 ⋅ 3 5 1 ⋅ 2 7 3
Combining Powers Combine the powers of 2: 2 1 + 3 2 + 5 1 + 7 3 ⋅ 3 5 1
Finding Common Denominator Find a common denominator for the exponents of 2. The common denominator is 105: 1 + 3 2 + 5 1 + 7 3 = 105 105 + 105 70 + 105 21 + 105 45 = 105 105 + 70 + 21 + 45 = 105 241
Simplified Expression Rewrite the expression as: 2 105 241 ⋅ 3 5 1
Final Result Therefore, the simplified expression is 2 105 241 ⋅ 3 5 1 .
Examples
Understanding how to simplify expressions with radicals and exponents is crucial in many fields, such as physics and engineering. For instance, when dealing with wave functions in quantum mechanics or analyzing signal processing, you often encounter expressions involving radicals and exponents. Simplifying these expressions allows for easier manipulation and interpretation of the results, leading to more efficient problem-solving and a deeper understanding of the underlying phenomena.
