Multiplication of Powers: Variables in the exponent of the power

$2^{2x+1}\cdot2^5\cdot2^{3x}=$

Since the bases are the same, the exponents can be added:

$2x+1+5+3x=5x+6$

$2^{5x+6}$

$4^{2y}\cdot4^{-5}\cdot4^{-y}\cdot4^6=$

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply the property for this problem:

$4^{2y}\cdot4^{-5}\cdot4^{-y}\cdot4^6= 4^{2y+(-5)+(-y)+6}=4^{2y-5-y+6}$We simplify the expression we got in the last step:

$4^{2y-5-y+6} =4^{y+1}$When we add similar terms in the exponent.

Therefore, the correct answer is option c.

$4^{y+1}$

$7^{2x+1}\cdot7^{-1}\cdot7^x=$

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply the property to our expression:

$7^{2x+1}\cdot7^{-1}\cdot7^x=7^{2x+1+(-1)+x}=7^{2x+1-1+x}$We simplify the expression we got in the last step:

$7^{2x+1-1+x}=7^{3x}$When we add similar terms in the exponent.

Therefore, the correct answer is option d.

$7^{3x}$

$3^x\cdot2^x\cdot3^{2x}=$

In this case we have 2 different bases, so we will add what can be added, that is, the exponents of $3$

$3^x\cdot2^x\cdot3^{2x}=2^x\cdot3^{3x}$

$3^{3x}\cdot2^x$

Solve for a:

$\frac{a^{3b}}{a^{2b}}\times a^b=$

$a^{2b}$

Solve the exercise:

$\frac{a^{2x}}{a^y}\times\frac{a^{2y}}{a^{-y}}=$

$a^{2(x+y)}$