$\frac{9 \frac{1}{3} \times 27 \frac{1}{2}}{3 \frac{-1}{6} \times 3 \frac{1}{3}}$
2) $\left(\frac{81}{16}\right)^{-\frac{3}{4}} \times\left(\frac{25}{9}\right)^{-\frac{3}{2}}$
Answer (2)
Convert mixed fractions to improper fractions.
Simplify the first expression by multiplying and dividing the fractions: 3 6 − 1 × 3 3 1 9 3 1 × 27 2 1 = 17 462 .
Simplify the second expression using exponent rules: ( 16 81 ) − 4 3 × ( 9 25 ) − 2 3 = 125 8 .
The simplified expressions are 17 462 and 125 8 .
Explanation
Problem Analysis We are given two expressions to simplify:
3 6 − 1 × 3 3 1 9 3 1 × 27 2 1
( 16 81 ) − 4 3 × ( 9 25 ) − 2 3
Converting Mixed Fractions Let's simplify the first expression. First, convert the mixed fractions to improper fractions: 9 3 1 = 3 9 × 3 + 1 = 3 28 27 2 1 = 2 27 × 2 + 1 = 2 55 3 6 − 1 = 6 3 × 6 − 1 = 6 17 3 3 1 = 3 3 × 3 + 1 = 3 10
Substituting Improper Fractions Now, substitute these improper fractions into the expression: 6 17 × 3 10 3 28 × 2 55 = 6 × 3 17 × 10 3 × 2 28 × 55 = 18 170 6 1540
Dividing Fractions To divide the fractions, we multiply by the reciprocal of the denominator: 18 170 6 1540 = 6 1540 × 170 18 = 6 × 170 1540 × 18 = 1020 27720
Simplifying the Fraction Simplify the fraction: 1020 27720 = 102 2772 = 51 1386 = 17 462 = 27 17 3 ≈ 27.176
Simplifying the Second Expression Now let's simplify the second expression: ( 16 81 ) − 4 3 × ( 9 25 ) − 2 3 Apply the negative exponent rule: a − n = a n 1 ( 16 81 ) − 4 3 = ( 81 16 ) 4 3 ( 9 25 ) − 2 3 = ( 25 9 ) 2 3
Rewriting the Bases Rewrite the bases as powers: 81 16 = ( 3 2 ) 4 25 9 = ( 5 3 ) 2
Applying the Power of a Power Rule Substitute these into the expression: ( ( 3 2 ) 4 ) 4 3 × ( ( 5 3 ) 2 ) 2 3 Apply the power of a power rule: ( a m ) n = a mn ( 3 2 ) 4 × 4 3 × ( 5 3 ) 2 × 2 3 = ( 3 2 ) 3 × ( 5 3 ) 3
Evaluating the Expression Evaluate the expression: ( 3 2 ) 3 × ( 5 3 ) 3 = 3 3 2 3 × 5 3 3 3 = 27 8 × 125 27 = 27 × 125 8 × 27 = 125 8 = 0.064
Final Answer Therefore, the simplified expressions are:
17 462 ≈ 27.176
125 8 = 0.064
Examples
Understanding how to simplify complex fractions and exponents is crucial in many fields, such as physics and engineering. For example, when calculating electrical circuits, you often encounter complex fractions involving resistance and impedance. Simplifying these fractions allows engineers to determine the overall behavior of the circuit and design it effectively. Similarly, in physics, exponential functions are used to model radioactive decay or population growth. Simplifying expressions with exponents helps scientists make accurate predictions and understand the underlying phenomena.
The simplified expressions are 17 462 and 125 8 . Both expressions were simplified step by step, using rules for fractions and exponents. The methods include converting mixed numbers to improper fractions and applying the negative exponent rule.
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