Substitute the values of \(\displaystyle a\) and \(\displaystyle b\).

\(\displaystyle \frac{15}{\sqrt{5}\cdot \sqrt{3}} = \frac{15}{\sqrt{15}}\)

Rationalize the denominator by multiplying the top and bottom with the denominator.  This will eliminate the radical in the denominator.

\(\displaystyle \frac{15}{\sqrt{15}}\cdot \frac{\sqrt{15}}{\sqrt{15}}=\frac{15\sqrt{15}}{15}\)

Cancel the integers.

The answer is:  \(\displaystyle \sqrt{15}\)

In order to solve this expression, we will need to substitute the assigned values into \(\displaystyle a\) and \(\displaystyle b\).

\(\displaystyle 3a^2-9b^3 = 3(6)^2-9(2)^3\)

Simplify the terms by order of operation.

\(\displaystyle 3(6)^2-9(2)^3 = 3(36)-9(8) = 108-72=36\)

The answer is:  \(\displaystyle 36\)

Substitute the values into the expression.

\(\displaystyle 2a^2-4b^3 = 2(5)^2-4(-3)^3\)

Simplify the terms by order of operations.

\(\displaystyle 2(5)^2-4(-3)^3 = 2(25) - 4(-27)\)

\(\displaystyle =50 + 108=158\)

The answer is:  \(\displaystyle 158\)

Substitute the values into the expression.

\(\displaystyle -(6)^{-2}-(-6)^{-2}\)

In order to evaluate this expression, we will need to rewrite the negative exponents into fractions.

\(\displaystyle -(6)^{-2}-(-6)^{-2}= -\frac{1}{6^2}-\frac{1}{(-6)^2}\)

Simplify the fractions.

\(\displaystyle -\frac{1}{36}-\frac{1}{36} =- \frac{2}{36}\)

Reduce this fraction.  

The answer is:  \(\displaystyle -\frac{1}{18}\)

Substitute the values in the expression.

\(\displaystyle \frac{b}{\sqrt a} = \frac{2}{\sqrt 3}\)

Rationalize the denominator by multiplying square root of three on the top and bottom of the fraction.

\(\displaystyle \frac{2}{\sqrt 3}\cdot \frac{\sqrt 3}{\sqrt 3}\)

Simplify the top and bottom.

The answer is:  \(\displaystyle \frac{2\sqrt{3}}{3}\)

The grocer wants the cost of the mixture to be less than the selling price. 

Selling price = 14.99

Find the cost of the mixture

1. cost of cashews in mix: \(\displaystyle 8.99 x\)

2. cost of almonds in mix: \(\displaystyle 6.99 y\)

3. cost of walnuts in mix: \(\displaystyle 10.99 z\)

The cost of the mixture = \(\displaystyle 8.99 x+6.99 y+10.99z\)

Cost of the mixture is less than Selling price

\(\displaystyle 8.99x+6.99y+10.99z< 14.99\)

If we write the number of pineapples ordered as \(\displaystyle p\), then \(\displaystyle 6p\) is the price of all the pineapples since each one costs \(\displaystyle \$6\). Now, it is alright if you have exactly \(\displaystyle \$400\) so it is an inclusive inequality, and you are drawing from an available amount of \(\displaystyle \$1000\), so we can write this inequality in words as

6 dollars * (number of pineapples) from 1000 dollars must be greater than or equal to 400 dollars. 

This can easily be converted to mathematical symbols as 

\(\displaystyle 1000-6p\geq 400\)

If Kirsten is represented by the variable \(\displaystyle x\), and Angie is represented by the variable \(\displaystyle y\), then the total amount Kirsten and Angie are spending can be represnted by the sum of \(\displaystyle x+y\). Since Angie and Kirsten only have \(\displaystyle \$200\) to spend on a couch,  \(\displaystyle x+y\) must be less than \(\displaystyle \$ 200\). Therefore, \(\displaystyle x+y< 200\). Kirsten only has \(\displaystyle \$ 75\) to spend on a couch. Therefore, \(\displaystyle x< 75\).

The first statement says that the total number of pants and shirts does not exceed 20. This means that the total can be 20 but nothing more which is represented by:

\(\displaystyle s+p\leq20\)

The second sentence says that the cost was greater than 50 dollars which is represented by:

\(\displaystyle c>50\)

To set up this inequality, break up the statement into parts.  Assume \(\displaystyle x\) is the number.

Twice a number:  \(\displaystyle 2x\)

Is less than: \(\displaystyle < \)

Product of the number and five:  \(\displaystyle 5x\)

Twice the product of the number and five:  \(\displaystyle 2(5x)\)

Connect the statement.

The answer is:  \(\displaystyle 2x< 2(5x)\)