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Evaluating expressions with exponents and nested brackets

Evaluating expressions with exponents and nested brackets

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Requirements

 

EVALUATING EXPRESSIONS WITH EXPONENTS AND NESTED BRACKETS

 

In this lesson, we will learn to simplify slightly more complicated algebraic expressions. For this, we need to know the order of operations as well as the order of brackets to be expanded.

Remember, in previous lessons, we learned the order of operations that we must follow in order to solve an expression. Let’s review them.

First of all, if there are any brackets in the expression, they need to be solved first. Second, we evaluate the powers or exponents. Third, we complete any division or multiplication. Lastly, we evaluate any addition or subtraction.

If there are nested brackets in the expression, the inner most brackets are solved first, then the outer brackets, and at the very end the outer most brackets are expanded. When you have brackets inside brackets, the best way to remember the order is think “inside to outside.” We must be careful to always follow the order of operations and the order of brackets in order to get the correct result of the expression.

Let’s practice some examples. Our first expression is

 

Before we do any simplification, we need to change the negative exponent into a positive exponent by taking the reciprocal of the term in the numerator. It should then look like this:

 

Then we distribute the exponent onto each factor in the brackets to get

 

 

Using the power rule of exponents and writing out 62 and 32 into 36 and 9 respectively, we have

 

.

 

Next step is we multiply the factors in the denominator. We simultaneously multiply the coefficients and the variables of the factors. Remember that in multiplying like variables, we use the product rule of exponents. It states that when like variables are multiplied, we retain the variable and add their powers. For unlike variables, we write them side by side. This makes the expression

 

 

 

which is simplified to

 

 We know that any variable raised to the power 0 is equal to 1, so  , making the expression

 

.

 We can’t have negative exponents in our answer, so we must convert the negative exponent of  into a positive one by taking its reciprocal. Your final answer should look like this:

 

Now let’s solve a simple example of solving nested brackets, that is, brackets inside brackets. Our first example on nested brackets is

.

Our very first step is to solve the innermost brackets and for that purpose, we will combine the like terms and rearrange the terms,

.

Now add the  within the brackets. Remember that when adding like terms, we just add their coefficients. So we now have the expression as

.

When we know that we cannot do any further simplification, we expand the brackets, keeping in mind that if there is minus sign outside the parenthesis, all the signs inside the brackets will be reversed.

But in this example, we have a plus sign outside the parenthesis, so no change will be done except to multiply the terms by 3 which makes the expression

Next step is we simplify the terms inside the square brackets. 4x and 9x are like terms so we combine them to have

 

We will now expand the brackets and multiply the terms by 6,

.

It’s as simple as that. You just need to be careful about possible minus signs while expanding the brackets.

Let’s practice a slightly more complex example,

. 

First of all, solve the innermost brackets and expand them. Be very careful when expanding brackets, because there is a minus sign before the brackets, we need to change the signs of the terms inside giving us

.

Next, we group like terms within the curly brackets,

 

and add the like terms which yields

Expanding the curly brackets gives us

Now we combine like terms inside the square brackets to have

.

Then we multiply the terms of the square brackets by the preceding 2,

.

Lastly, we group like terms as

 

then finally we combine like terms to get the answer 

.

  It’s very easy to solve the nested brackets if you follow the whole procedure carefully step by step. Let’s work out another example,

 .

 First of all, solve the innermost brackets and expand them. We add -5b and b to make the expression

  

 Now there is a negative sign before the innermost brackets. Removing the brackets means changing all the signs inside the brackets. So we have

 .

 Now we can rearrange the nested expression to make solving it easier. It becomes

 .

We then combine like terms within the nested bracket expression, which is now

 .

 To remove the final set of nested brackets, we multiply each factor by 3 and we get

  

 Then we rearrange the factors to make operations on them easier like this:

.

Then we can reduce the bracketed expression down to

 

We can remove the last set of brackets in this expression by changing all the signs of the terms inside the square brackets,

.

We rearrange the factors to simplify the mathematical operations,

.

And finally, we simplify this expression by adding like terms 5a and 9b to get the result

.

Excellent! In this lesson, we learned how to simplify slightly more complicated algebraic expressions. We also learned the order of operations, as well as the order of brackets to be expanded. With a bit of practice, you will quickly see how nesting can make complicated ideas into simple algebraic expressions that you can then solve.