When multiplying expressions with the same variable, combine terms by adding the exponents, while leaving the variable unchanged. For this problem, we do that by adding (1/2)+(1/2), to get a new exponent of 1:

$\displaystyle x^{\frac{1}{2}}\cdot x^{\frac{1}{2}}=x ^{\frac{1}{2}+\frac{1}{2}}=x^{1}=x$

When multiplying expressions with the same variable, combine terms by adding the exponents, while leaving the variable unchanged. For this problem, we do that by adding 9+2, to get a new exponent of 11:

$\displaystyle x^{9}\cdot x^{2}=x^{9+2}=x^{11}$

When multiplying expressions with the same variable, combine terms by adding the exponents, while leaving the variable unchanged. For this problem, we do that by adding (1/5)+(3/5), to get a new exponent of (4/5):

$\displaystyle x^{\frac{1}{5}}\cdot x^{\frac{3}{5}}=x ^{\frac{1}{5}+\frac{3}{5}}=x^{\frac{4}{5}}$

When multiplying expressions with the same variable, combine terms by adding the exponents, while leaving the variable unchanged. For this problem, we do that by adding 3+9, to get a new exponent of 12:

$\displaystyle x^{3}\cdot x^{9}=x ^{3+9}=x^{12}$

When multiplying expressions with the same variable, combine terms by adding the exponents, while leaving the variable unchanged. For this problem, we do that by adding 2+11, to get a new exponent of 13:

$\displaystyle x^{2}\cdot x^{11}=x ^{2+11}=x^{13}$

When multiplying expressions with the same variable, combine terms by adding the exponents, while leaving the variable unchanged. For this problem, we do that by adding 5+5, to get a new exponent of 10:

$\displaystyle x^{5}\cdot x^{5}=x ^{5+5}=x^{10}$

When multiplying exponents, we just add the exponents while keeping the base the same.

$\displaystyle 4^7*4^{18}=4^{7+18}=4^{25}$

When multiplying exponents with different bases but the same exponent, you multiply the bases and keep the exponents the same.

$\displaystyle 7^8*5^8=(7*5)^8=35^8$

Set $\displaystyle a =100$ and $\displaystyle b= 2$ in the expression in the definition, then simplify the exponent:

$\displaystyle a \bigstar b = a ^{2+ b }$

$\displaystyle 100 \bigstar (-2)= 100 ^{2+ (-2)}$

$\displaystyle 100 \bigstar (-2)= 100 ^{0}$

Any nonzero number raised to the power of 0 is equal to 1, so 

$\displaystyle 100 \bigstar (-2)= 1$.

Set $\displaystyle a =100$ and $\displaystyle b= \frac{1}{2}$ in the expression in the definition:

$\displaystyle a \bigstar b =\left ( \frac{1}{a} \right )^ {\frac{1}{b}}$

$\displaystyle 100 \bigstar \frac{1}{2 }=\left ( \frac{1}{100} \right )^ {\frac{1}{ \frac{1}{2}}}$

The exponent $\displaystyle \frac{1}{ \frac{1}{2}}$ simplifies as follows:

$\displaystyle \frac{1}{ \frac{1}{2}} = 1 \div \frac{1}{2} = 1 \cdot 2 = 2$, so

$\displaystyle 100 \bigstar \frac{1}{2 }=\left ( \frac{1}{100} \right )^ {2}= \frac{1}{100} \cdot \frac{1}{100} = \frac{1}{10,000}$