Rewrite using only positive, rational exponents.
$6 a^{\frac{3}{2}} b^{-\frac{4}{5}} c^{\frac{5}{3}}$
Answer (1)
Identify the term with the negative exponent: b − 5 4 .
Rewrite the term with a positive exponent by moving it to the denominator: b − 5 4 = b 5 4 1 .
Substitute the rewritten term back into the original expression.
The final expression with only positive, rational exponents is: b 5 4 6 a 2 3 c 3 5 .
Explanation
Understanding the Problem We are given the expression 6 a 2 3 b − 5 4 c 3 5 and asked to rewrite it using only positive, rational exponents. This means we need to deal with the negative exponent on the variable b .
Handling the Negative Exponent To rewrite the expression with positive exponents, we recall that x − n = x n 1 . Applying this to our expression, we have b − 5 4 = b 5 4 1 .
Rewriting the Expression Now we substitute this back into the original expression: 6 a 2 3 b − 5 4 c 3 5 = 6 a 2 3 ⋅ b 5 4 1 ⋅ c 3 5 = b 5 4 6 a 2 3 c 3 5 .
Final Answer Therefore, the expression rewritten with only positive, rational exponents is b 5 4 6 a 2 3 c 3 5 .
Examples
Understanding how to manipulate exponents is crucial in various scientific fields. For instance, in physics, when dealing with gravitational force, the force is inversely proportional to the square of the distance ( r ) between two objects, expressed as F = G r 2 m 1 m 2 = G m 1 m 2 r − 2 . Rewriting expressions with positive exponents helps in simplifying calculations and understanding relationships between variables. Similarly, in finance, compound interest formulas involve exponents, and rewriting them can aid in analyzing growth rates and investment returns.
