\( 7\sqrt{9} + 3 - \sqrt{64} \)

First calculate the two square root terms.

\( 7(3) + 3 - 8 \)

Now we look closely at the expression. The parentheses contain only one term, so we can remove them.

\( 7*3 + 3 - 8 \)

Now we do the multiplication since it is a higher priority than the addition and subtraction.

\( 21 + 3 - 8 \)

Now we have only addition and subtraction, so we work left to right.

\( 24 - 8 \)

\( 16 \)

\( 8 - 2[ 3 - 4\sqrt{50-14} + 5] \)

First, we evaluate the square root term.

\( 8 - 2[ 3 - 4\sqrt{36} + 5] \)

\( 8 - 2[ 3 - 4*6 + 5] \)

Next, we focus on the expression in the square brackets. Square brackets are sometimes used instead of parentheses and they have the same priority as parentheses. Inside the brackets, we have addition, subtraction and multiplication. Multiplication is the highest priority, so we perform that operation first.

\( 8 - 2[ 3 - 24 + 5] \)

Now we can evaluate the expression in the brackets, left to right.

\( 8 - 2[ -21 + 5] \)

\( 8 - 2[ -16 ] \)

Now take the negative sign outside the brackets.

\( 8 + 2[ 16 ] \)

Since we have only term inside the brackets, they are no longer needed.

\( 8 + 2*16 \)

Multiplication is a higher priority than addition, so we perform the multiplication operation.

\( 8+32 = 40 \)

\( 42 \div 2\sqrt{ 4^2 \cdot 5 - 2 (8.7+6.8) } \)

First, we start inside the square root. Inside we have a power, multiplication, addition and subtraction and some parentheses. We start with the parentheses.

\( 42 \div 2\sqrt{ 4^2 \cdot 5 - 2 (15.5) } \)

Since we have only one single term inside the parentheses, they are no longer necessary.

\( 42 \div 2\sqrt{ 4^2 \cdot 5 - 2 \cdot 15.5 } \)

So now, the next priority is the power. So we evaluate \(4^2\). Notice that we do NOT multiply the power of 2 by 5 first. That's because, the power is a higher priority than the multiplication.

\( 42 \div 2\sqrt{ 16 \cdot 5 - 2 \cdot 15.5 } \)

Now we have multiplication and subtraction inside the square root. So we do the multiplication first, left to right.

\( 42 \div 2\sqrt{ 80 - 2 \cdot 15.5 } \)

\( 42 \div 2\sqrt{ 80 - 31 } \)

Now do the subtraction inside the square root.

\( 42 \div 2\sqrt{ 49 } \)

And take the square root.

\( 42 \div 2 \cdot 7 \)

Now we are only left with multiplication and division. So we work left to right.

\( 21 \cdot 7 = 147 \)

You may be asking yourself, why do we start with the square root? It is not in parentheses. Well, actually it is. You can write a square root \(\sqrt{x}\) as \( (x)^{1/2} \). So this is an expression \(x\) inside parentheses with a power \(1/2\). So that is why we started with the square root.

\(\displaystyle{ \frac{ [(8+5)(6-2)^2] - (4 \cdot 17 \div 2) }{(24 \div 2) \div 3} }\)

To get started with this one, we need to start with expressions in the most inside parentheses. We will also work separately with numerator and denominator until we get two individual terms to divide. Let's start with the most inside parentheses in the numerator, \((8+5=13)\) and \((6-2=4)\).

\(\displaystyle{ \frac{ [(13)(4)^2] - (4 \cdot 17 \div 2) }{(24 \div 2) \div 3} }\)

So now we can remove the parentheses from the \( (13) \) and from the \( (4) \).

\(\displaystyle{ \frac{ [13 \cdot 4^2] - (4 \cdot 17 \div 2) }{(24 \div 2) \div 3} }\)

Staying inside the same set of brackets, we have multiplication and a power. The power has a higher priority, so we do that first.

\(\displaystyle{ \frac{ [13 \cdot 16] - (4 \cdot 17 \div 2) }{(24 \div 2) \div 3} }\)

So now were are left with just one operation inside the brackets, so we multiply those values and drop the brackets.

\(\displaystyle{ \frac{ 208 - (4 \cdot 17 \div 2) }{(24 \div 2) \div 3} }\)

Okay, so staying in the numerator, we will work with the expression in the remaining parentheses. We have only multiplication and division, so we work left to right.

\(\displaystyle{ \frac{ 208 - (68 \div 2) }{(24 \div 2) \div 3} = \frac{ 208 - (34) }{(24 \div 2) \div 3} = \frac{ 208 - 34 }{(24 \div 2) \div 3} }\)

Finishing off the numerator we perform the subtraction.

\(\displaystyle{ \frac{ 174 }{(24 \div 2) \div 3} }\)

Now we need to move to the denominator. We will first perform the operation inside the parentheses and then drop the parentheses.

\(\displaystyle{ \frac{ 174 }{(12) \div 3} = \frac{ 174 }{12 \div 3} }\)

Next, we perform the last operation in the denominator and finish the problem.

\(\displaystyle{ \frac{ 174 }{4} = 43.5 }\)

Notice how we focused on very small parts of an otherwise complicated expression. This is a very useful concept when working algebra problems.