We are asked to evaluate each expression for the given values.

To evaluate an algebraic expression we first substitute the given values for

the variables and then we evaluate or simplify the given expression using the

order of operations. For number one we're given the expression 2x - 5.

We're asked to evaluate this when x equals negative 6. For the first step we substitute

negative 6 for X and because two X means 2 times X, performing substitution gives

us the expression 2 times negative 6 minus five.

Following the order of operations, we multiply before subtracting. 2 times

negative 6 equals negative 12.

This simplifies to negative 12 minus 5. Negative 12 minus 5 is equal to negative 17. The expression

is equal to negative 17 when x equals negative 6.

Now if it's helpful we could have written this difference as a sum.

Subtracting positive 5 is equivalent to adding negative 5 and therefore negative

12 minus 5 is equivalent to negative 12 plus negative 5, which does give us negative 17.

For number two, we have the expression four divided by the quantity 5t minus

four, which we want to evaluate when T equals four.

We substitute four for t, which gives us the expression four divided by the

quantity 5 times 4 minus 4.

We need to be careful here. We can not simplify these fours here or these fours

because we cannot simplify across addition or subtraction.

When we have a fraction we treat this as if we have grouping symbols around the

numerator and denominator. Therefore the next step is to simplify the

denominator. In the denominator we have multiplication and subtraction.

We multiply first. So we have 4 divided by the quantity 5 times 4 is equal to 20.

The denominator simplifies to the quantity 20 minus4. 20 minus 4 is equal to 16.

Therefore this simplifies to four sixteenths, which does simplify because 4 and 16 share a

common factor of 4. To simplify we divide the numerator and denominator by 4. 4 divided by 4

equals 1. 16 divided 4 is equal to four. The expression is equal to one fourth when T equals 4.

The next expression is two-thirds times

pi r squared h, which we want to evaluate when

r equals 3 and h equals seven. We substitute 3 for r and 7 for h.

This gives us the expression two-thirds times pi times the square of three times

seven.

The first step is to simplify the exponents. 3 squared is equal to 3 times

3 which equals nine. We have two thirds pi x 9 x 7. Now we multiply.

But before multiplying we can simplify because three and nine share a common

factor 3. As as a fraction nine times sevenhas a denominator of 1. Now

simplifying ,there's one three in three and three threes in nine. Multiplying we

have 2 times pi times 3 times 7 and because 2 times 3 times 7 is equal to 42,

this product as 42 pi.

The given expression is equal to 42 pi,

when r equals 3 and h equals seven. We often use 3.14 as an approximation for

pi

but because we want the exact value of the expression, we leave this in terms of

pi. The next expression is a squared plus 2 AB plus B squared,

which we went to evaluate when A equals negative 3 and B equals 6.

Performing the substitution a squared would be the square of negative 3.

We do need parentheses around the negative 3 because we went two factors

of negative 3.

Then we have plus 2 times A times B which is 2 times negative 3 times 6 plus

B squared which is the square of six.

The next step is to simplify the exponents. Being careful here

the square negative 3 means negative 3 times in 3 which is positive 9.

Then we have plus 2 times negative 3 times 6 plus 6 squared is equal to positive 36.

Next we multiply. We have nine plus 2 times negative 3 times six equals

negative 36.

So we have plus negative 36, which we also write as just minus 36 and then plus 36.

The last step is to add notice here we're adding opposites so this sum is 0.

This is 9 plus 0 which equals nine or adding from left to right

nine plus negative 36 equals negative 27 of 27 plus 36 equals positive 9.

The last expression is the square root of 5 A squared BC.

We want to evaluate this when a equals to two, B equals 5 and C equals 1.

Performing the substitution gives us the expression the square root of five times

the square of 2 times 5 times 1.

Simplifying under the square root we first simplify the exponents. 2 squared

is equal to 2 times 2 which equals four.

We have the square root of 5 times 4 times 5.

we don't need the times 1 but let's go and leave it there. Next we multiply under

the square root.

5 times 4 times 5 times 1 is equal to 100.

The square root of 100 simplifies perfectly to 10. The square root of 100 is equal to 10 because

10 squared equals 100.

I hope you found this helpful.