Write with fractional exponents. [tex]65\sqrt{x^2 y}[/tex]
Answer (2)
The expression 65 x 2 y can be rewritten using fractional exponents as 65 x y 2 1 . This is achieved by converting the square root into a fractional exponent and applying the power of a product rule. Finally, we simplify the terms to reach the final expression.
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Rewrite the square root as a fractional exponent: x 2 y = ( x 2 y ) 2 1 .
Apply the power of a product rule: ( x 2 y ) 2 1 = ( x 2 ) 2 1 y 2 1 .
Simplify the exponents: ( x 2 ) 2 1 y 2 1 = x 2 ⋅ 2 1 y 2 1 = x y 2 1 .
Multiply by the constant: 65 x y 2 1 . The final answer is 65 x y 2 1 .
Explanation
Understanding the Problem We need to rewrite the expression $65
\sqrt{x^2 y}$ using fractional exponents. This involves understanding how radicals relate to fractional powers.
Converting the Square Root to a Fractional Exponent The square root can be expressed as a fractional exponent of 2 1 . Therefore, we can rewrite the expression as:
x 2 y = ( x 2 y ) 2 1 .
Applying the Power of a Product Rule Now, we apply the power of a product rule, which states that ( ab ) n = a n b n . Applying this rule, we get:
( x 2 y ) 2 1 = ( x 2 ) 2 1 y 2 1 .
Simplifying the Exponents Next, we simplify the exponents. Recall that $(x^a)^b = x^{a
\cdot b}$. Thus, we have:
( x 2 ) 2 1 y 2 1 = x 2 ⋅ 2 1 y 2 1 = x 1 y 2 1 = x y 2 1 .
Multiplying by the Constant Finally, we multiply by the constant 65 to get the final expression:
65 x y 2 1 .
Final Answer Therefore, the expression $65
\sqrt{x^2 y}$ written with fractional exponents is 65 x y 2 1 .
Examples
Fractional exponents are useful in various scientific and engineering calculations. For example, in physics, the period of a pendulum can be expressed using a fractional exponent. If the length of the pendulum is l and the acceleration due to gravity is g , then the period T is given by T = 2 π g l = 2 π ( g l ) 2 1 . Similarly, in finance, compound interest calculations often involve fractional exponents to determine the growth of investments over time.
