Simplify [tex]$16^{-\frac{1}{2}} \times 4^{\frac{1}{2}} \times 27^{\frac{1}{3}}$[/tex]
Answer (2)
Rewrite each term with its prime factorization.
Apply the power of a power rule to simplify the exponents: 2 − 2 × 2 1 × 3 1 .
Rewrite 2 − 2 as 2 2 1 = 4 1 .
Simplify the expression to get the final result: 2 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 1 6 − 2 1 × 4 2 1 × 2 7 3 1 . This involves working with fractional exponents, which represent roots and powers.
Rewriting with Prime Factorization First, let's rewrite each term using its prime factorization and exponent rules. We know that 16 = 2 4 , 4 = 2 2 , and 27 = 3 3 . Substituting these into the expression, we get: ( 2 4 ) − 2 1 × ( 2 2 ) 2 1 × ( 3 3 ) 3 1 Now, we apply the power of a power rule, which states that ( a m ) n = a mn . Applying this rule to each term, we have: 2 4 ( − 2 1 ) × 2 2 ( 2 1 ) × 3 3 ( 3 1 )
Simplifying Exponents Next, we simplify the exponents: 2 − 2 × 2 1 × 3 1 Recall that a − n = a n 1 . Therefore, 2 − 2 = 2 2 1 = 4 1 . So the expression becomes: 4 1 × 2 × 3
Final Calculation Now, we multiply the numbers: 4 1 × 2 × 3 = 4 6 Finally, we simplify the fraction: 4 6 = 2 3 So, the simplified expression is 2 3 or 1.5.
Conclusion Therefore, the simplified form of the given expression is 2 3 .
Examples
Fractional exponents are used in various fields, such as physics and engineering, to model relationships between quantities. For example, the period of a pendulum is proportional to the square root of its length, which can be expressed using a fractional exponent. Simplifying expressions with fractional exponents helps in understanding and calculating these relationships more efficiently. In finance, compound interest calculations often involve fractional exponents when dealing with time periods that are not whole years.
The expression 1 6 − 2 1 × 4 2 1 × 2 7 3 1 simplifies to 2 3 . This was done by rewriting each number using prime factorization and applying exponent rules systematically. The final answer is 2 3 .
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