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question 10
which of the following is equal to ( 18x^{13/4} + 30x^{9/4} )?
○ ( 48x^{11/2} )
○ nothing in this list is correct.
○ ( 6x^{1/4}(3x^{13} + 5x^9) )
○ ( 6x^{9/4}(3x + 5) )

Explanation:

Step1: Find the GCF of coefficients and exponents

First, find the greatest common factor (GCF) of the coefficients 18 and 30. The GCF of 18 and 30 is 6. Then, for the variable part \(x^{13/4}\) and \(x^{9/4}\), the GCF (in terms of exponents) is the smaller exponent, which is \(x^{9/4}\) (since when factoring, we take the lowest power of \(x\) present).

Step2: Factor out the GCF

Factor out \(6x^{9/4}\) from \(18x^{13/4}+30x^{9/4}\).

• For the first term: \(\frac{18x^{13/4}}{6x^{9/4}} = 3x^{(13/4 - 9/4)} = 3x^{4/4}=3x\)
• For the second term: \(\frac{30x^{9/4}}{6x^{9/4}} = 5\)

So, factoring out \(6x^{9/4}\) gives \(6x^{9/4}(3x + 5)\).

Step3: Check other options (optional but for verification)

• Option 1: \(48x^{11/2}\) is adding the coefficients and averaging exponents, which is incorrect for factoring.
• Option 2: "Nothing in this list..." is wrong as we found a correct factorization.
• Option 3: If we expand \(6x^{1/4}(3x^{13}+5x^{9})\), we get \(18x^{13 + 1/4}+30x^{9+1/4}=18x^{53/4}+30x^{37/4}\), which is not the original expression.

Answer:

\(6x^{9/4}(3x + 5)\) (the last option: \(6x^{9/4}(3x + 5)\))