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Math  /  Algebra

Question(C) 12x1218x−52\frac{12 x^{\frac{1}{2}}}{18 x^{-\frac{5}{2}}}

Studdy Solution

STEP 1

1. We are given a rational expression involving exponents.
2. The expression can be simplified by applying the properties of exponents.

STEP 2

1. Simplify the coefficients in the expression.
2. Simplify the exponents using the properties of exponents.

STEP 3

Simplify the coefficients in the expression 1218\frac{12}{18}.
1218=23 \frac{12}{18} = \frac{2}{3}

STEP 4

Simplify the expression by applying the properties of exponents. Specifically, use the rule aman=am−n\frac{a^m}{a^n} = a^{m-n}.
x12x−52=x12−(−52) \frac{x^{\frac{1}{2}}}{x^{-\frac{5}{2}}} = x^{\frac{1}{2} - (-\frac{5}{2})}

STEP 5

Calculate the exponent:
12−(−52)=12+52=62=3 \frac{1}{2} - (-\frac{5}{2}) = \frac{1}{2} + \frac{5}{2} = \frac{6}{2} = 3
So, the expression simplifies to:
x3 x^3

STEP 6

Combine the simplified coefficient and the simplified exponent expression:
23×x3=2x33 \frac{2}{3} \times x^3 = \frac{2x^3}{3}
The simplified expression is:
2x33 \boxed{\frac{2x^3}{3}}

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