Grouping Symbols (Level 2). Alright
like always let's jump straight into some examples. Simplify the following expressions,
so here what we want to do first is follow the rules of
simplifying expressions without grouping symbols
In this case we first need to take care of multiplication or division operators
and then after that we need to take care of any subtraction
and addition operators.
so scanning the equation from left to right there is an addition operator
so we skip it for now
and right next to it there is a multiplication operator
so according to our rules we take care of that first.
So we simplify the expression 4 times 2, which simplifies to 8,
and then we add 8 to that which reduces to 16.
Alright let's try the next one.
Here we have a parenthesis we need to simplify the expression within the grouping symbol
first add 8 and 2 which simplifies to 10 and then here we will multiply 10
by 4 which gives us a final answer of 40.
Alright let's try the next one, so here once again we dont have any grouping symbols
so we first need to take care of any multiplication or division operators first
scanning the equation from left to right we first spot a division operator
so we take care of that first, so 10 divided by 5 is equal to 2,
and then we add 5 to that.
Let's try the next one.
So here it looks like that this expression is similar to the last one
but the only difference is that
we have a grouping symbol from the start
so like always we simplify the expression within the grouping symbol first,
so adding 10 and 5 gives us 15 and now we take 15
and divided by 5 which reduces to 3.
Alright let's try the next one
so this expression involves a fraction bar
so the first thing we need to do is simplify the numerator and then simplify the denominator
and in the end divide the simplified numerator over the simplified denominator.
let's work with the numerator first
so here we first need to take care of the product 5 times 3
which reduces to 15 and then we add 9 to that which is equal to 24.
In the denominator we add 7 and 1 which simplifies to 8,
and now we take 24 and divide it by 8 which reduces to 3.
Alright let's try the next one.
Once again we have a fraction bar so we take care of the numerator and the denominator first.
Within the numerator we first subtract 12 and 8 which reduces to 4,
and then multiply that by 3 which reduces to 12 in the denominator we first need to take care of the product
5 times 2 which is equal to 10 and then subtract 4 to that which reduces to 6.
and lastly we take 12 and divide it by 6 which is equal to 2.
Alright so now let's practice evaluating expressions.
Evaluate the following expressions if t is equal to 6, x is equal to 3
y is equal to 4 and z is equal to 5.
The way you want to approach these problems is by replacing or substituting each variable
with the corresponding value of the variable typically with a parenthesis.
Make sure you use the parenthesis when substituting values this will be very important
in your future studies of algebra.
So simplify the expression 2x plus 7
So like we want to substitute the variable with the corresponding value
in this case x is equal to 3.
So we use parenthesis to substitute x with 3,
and now we simplify the expression you first take care of the product 3 times 2
which reduces to 6 and then we add 7 to that which reduces to 13.
Alright let's try the next one.
The quantity 18 minus 4 times t.
So we fist simplify the expression within the parenthesis
and that reduces to 14
and now we have the expression 14 times t in this problem t is equal to 6.
so we substitute t using parenthesis with 6.
and now we have 6 times 14
which simplifies to 84. Alright let's try the next one.
The expression 4 times x times z or 4xz
plus 3 times y or 3y.
So we first need to substitute each variable with its appropriate value
x is equal to 3, z is equal to 5,
and y is equal to 4. So now we simplify any products or divisions first.
We simplify the expression 5 times 3 times 4 first.
Which simplifies to 60,
and then we simplify the expression 4 times 3.
which it simplifies to 12,
and finally we add these two quantities which simplifies to 72.
The quantity y times y plus z times z
Once again we have pure variables so we have a purely variable expression.
We substitute each variable with its appropriate value
y is equal to 4 and z is equal to 5.
So now we need to pay close attention to the grouping symbols, in this case the bracket.
You first need to take care of the expression within the bracket,
focusing on the expression within the bracket we see that we first need to take care of the product,
4 times 4, which reduces to 16 and now we add 5 to that which reduces to 21,
and finally we take 21 and multiply it by 5 which simplifies to 105.
Alright, try the following one.
The expression 2 times the quantity y plus x over the quantity 2y plus x.
Once again we need to substitute each variable
using parenthesis with the appropriate value
y is equal to 4 and x is equal to 3.
So now here we have a grouping simple specifically a fraction bar
we first need to take care of the expression within the parenthesis in the numerator
we first add the numbers 4 and 3 which simplifies to 7
and then multiply that by 2 which is equal to 14.
Next we take care of the denominator
we first need to multiply 4 and 2 which reduces to 8.
and then add 3 to that which reduces to 11,
and finally the final expression is 14 over 11.
For the most part leave your answer as an improper fraction
there's really no need to simplify it as a decimal
but if you like 14 over 11 is approximately equal to 1.27 after rounding
to the nearest hundredth there's no need to write the answer as a decimal.
So how about the folowing expression.
3 times the quantity z plus 5 times the quantity 2y minus x.
So this is a perfect example of an expression that contains nested grouping symbols.
Notice that we have the brackets and within the brackets we have an expression with parenthesis,
we work from the innermost grouping symbol to the outer most grouping symbol,
so in this case we first need to take care of the expression 2y minus x.
We first substitute each expression and as always with parenthesis
z is equal to 5, y is equal to 4 and x is equal to 3.
So now we need to take care of the quantity within the parenthesis first
we first multiply 4 times 2 which reduces to 8 and subtract 3 from that
which simplifies to 5 and now we simplify the expression within the brackets
we first need to take care of the expression 5 times 5 which simplifies to 25,
and then add 5 to that which reduces to 30
and then we take 30 and multiply that by 3
which simplifies to 90. Let's try a different type of problem.
Fill in the blank by using either an equal sign or a does not equal sign to make a true statement.
In these types of problems what you want to do first is simplify
the left hand side of the expression and the right hand side of the expression
so you first simplify the expression on the left hand side
in order to do that we need to simplify the numerator and the denominator.
So we first add the quantity 23 and 19 which simplifies to 42
and then simplify the denominator which simplifies to 14,
and then take 42 and divide it by 14 which simplifies to 3.
Now the right hand side contains no grouping symbols so we need to follow the rules
to simplify expressions without grouping symbols, you first need to divide
54 and 9 which simplifies to 6 and then subtract 3 from that,
which simplifies to 3. So now that we have the left hand side and the right hand side simplified
in order to make this a true statement we need to substitute the blank with an equal sign.
Let's try the next one
Once again we need to simplify the left hand side
we have once again nested grouping symbols
so we need to work the inner most grouping symbol in this case the parenthesis.
So first we add 6 and 3 which simplifies to 9
we work with the quantity within the bracket
so we take 36 and divide it by 9 which simplifies to 4,
and finally take 4 and multiply it by 3
which simplifies to 12. The right hand side also contains nested grouping symbols.
We work with the innermost grouping symbol first
so we take 36 and divide it by 3 which simplifies to 12,
and then we add 6 to that which simplifies to 18,
and finally we take 20 and subtract 18 which simplifies to 2
so now we have 12 on one side and 2 on the other side
the only way we can make this a true statement is by substituting the blank with a does not equal symbol.
The whole purpose of this problem is to get a true statement.
Alright let's try a distinct type of problem. Insert grouping symbols in the numerical expression
so that its value is equal to, a) 7, b) 17, and c) 25. Let's try a) first.
So in order to make this numerical expression equal to 7
the only way you could get that is by grouping the quantity 5 times 3 plus 6,
so by grouping the expression 5 times 3 plus 6
we would have to work with the product first
which simplifies to 15 and then add 6 to that which is equal to 21.
and finally we take 21 and divide it by 3 which simplifies to 7.
Alright so let's try b). In order to obtain 17 from this expression
notice that we don't actually have to write any grouping symbols
we can just follow the rules of simplifying numerical expressions without any grouping symbols
so in order to simplify this expression using the rules
we first need to take care of any multiplication or division operators
in this case we first take care of 5 times 3 which simplifies to 15
and then we take care of 6 divided by 3 which simplifies to 2.
So 15 plus 2 gives us 17.
In this particular example there is no need to insert any grouping symbols to give us a value of 17.
Alright so let's try the last one.
How can we group the numerical expression so that it gives us a final answer of 25?
The only way that we can get 25 is by grouping
the expressions 3 plus 6 divided by 3.
Doing so we would have to take care of the division sign first within the parenthesis,
so we take 6 divided by 3 which is equal to 2 and then we add 3 to that which is equal to 5
and finally 5 times 5 is equal to 25.
If you still don't feel confident I suggest you practice some more before you continue.
In our next video we will go over the basic structures of equations.