Question

3
which of the following is equivalent to ( 3y^{-\frac{1}{2}} ) for all ( y > 0 )?
a) ( sqrt{3y} )
b) ( sqrt{9y} )
c) ( sqrt{\frac{3}{y}} )
d) ( sqrt{\frac{9}{y}} )

4
which expression is equivalent to ( 4^{m + 2} )?
a) ( 16^m )
b) ( 16 + 4^m )
c) ( 8(4^m) )
d) ( 16(4^m) )

7
which expression is equivalent to ( x^{\frac{2a}{b}} ) for all positive values of ( x ), where ( a ) and ( b ) are positive integers?
a) ( sqrtb{ax^2} )
b) ( sqrtb{x^{2a}} )
c) ( sqrtb{x^{a + 2}} )
d) ( sqrt2a{x^b} )

8
if ( \frac{9^7}{sqrt4{9^{10}}} = 9^{6t} ), what is the value of ( t )?

Explanation:

Step1: Rewrite exponent as radical

$3y^{\frac{1}{2}} = 3\sqrt{y}$

Step2: Square the coefficient inside radical

$3\sqrt{y} = \sqrt{9}\sqrt{y} = \sqrt{9y}$

Step1: Split exponent using product rule

$4^{m+2} = 4^m \times 4^2$

Step2: Calculate $4^2$ and simplify

$4^m \times 16 = 16(4^m)$

Step1: Apply exponent to radical rule

$x^{\frac{2a}{b}} = \sqrt[b]{x^{2a}}$

Step1: Rewrite root as exponent

$\frac{9^7}{\sqrt[4]{9^{10}}} = \frac{9^7}{9^{\frac{10}{4}}}$

Step2: Simplify exponent fraction

$\frac{10}{4} = \frac{5}{2}$

Step3: Subtract exponents (same base)

$9^{7 - \frac{5}{2}} = 9^{\frac{14}{2} - \frac{5}{2}} = 9^{\frac{9}{2}}$

Step4: Set equal to $9^{6t}$ and solve

$\frac{9}{2} = 6t \implies t = \frac{9}{2 \times 6} = \frac{3}{4}$

Answer:

B) $\sqrt{9y}$

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