Order of Operations: Exponents

In the event that we have an exercise with parentheses that are within other parentheses, we will first solve the inner parentheses and then move on to the outer parentheses.

$2 + 5 \cdot 4^2 \cdot (3-1) =$

First, we perform the operations inside the parentheses. Once done, we obtain the following:
$2 + 5 \cdot 4^2 \cdot 2 =$

Note that we have a combined operation involving powers, multiplications, and additions, so we proceed to solve the power.

Once done, we obtain:
$2+5 \cdot16 \cdot 2 =$

Now it's time to solve the multiplications (remember: from left to right):
$​​2+80\cdot2=$,$2+160=​​$

Lastly, we add:
$2+160=162$

Now we will do the same exercise, but with a small variation.

$2 + 5 \cdot 4^2 \cdot (3^2-1) =$

First, we must solve the operation inside the parentheses, where there is a power and a subtraction, so following the order of operations we calculate the power and then the subtraction.

$2 + 5 \cdot 4^2 \cdot (3^2-1) =$

$2 + 5 \cdot 4^2 \cdot (9-1) =$

Now we can proceed with solving the exercise just as we have been doing so far.

$2 + 5 \cdot 4^2 \cdot 8 =$

$2 + 5 \cdot 16 \cdot 8 =$

$2 + 80 \cdot 8 =$

$2 + 640 = 642$

$(23+4^2)+8\cdot(9^2-1)=$

We solve the operations within each parenthesis, applying the order of operations within them.

$(23+16)+8\cdot(81-1)=$

Now is the time to solve the multiplications (we remember: from left to right):

$39+8\cdot80=$

$39+640=$

Lastly, we add:
$39+640=679$

Remember that the order of operations will always be the same, even when combined operations with fractions appear.

$4+2^2=$

First, the powers

$4+4=$

Lastly, we add

$4+4=8$

$4+2+5^2=$

First, the powers

$4+2+25=$

Lastly, we add

$4+2+25=31$

$5+5-5^2+4^2=$

First, the powers

$5+5-25+16=$

Lastly, we add

$5+5-25+16=1$

• $3 \cdot3+3^2=$
• $(3+1)^2-(4+1)=$
• $10:2-2^2=$
• $100:5^2+3^2=$
• $5^3:5^2\cdot2^3=$
• $0:2^2\cdot1^{10}+3$
• $({1\over4})^2+{1\over16}=$
• $({1\over2})^2+({1\over3})^2+{1\over4}=$
• $8-3^2:3=$
• $(2+1 \cdot2)^2=$
• $(20-3 \cdot2^2)^2=$
• $(15+9:3-4^2)^2=$
• $[(4-2^2)]^3=$
• $5+8^2=$
• $2^6+3=$
• $12-3^2=$
• $22-3^4=$
• $({1\over3})^2\cdot60=$
• $7-8 \cdot 2-3^2=$
• $25\cdot[({1\over2})^2+2^2]=$
• $27.5+1.5^3\cdot6=$
• $0.2^2\cdot5=$
• $(6-6)\cdot2^2=$
• $1+20^2\cdot{1\over5}=$

If you're interested in this article, you might also be interested in the following articles:

• The Order of Operations
• Basic Order of Operations: Addition, Subtraction, Multiplication, and Division
• Basic Order of Operations: Roots
• The 0 and 1 in the Order of Operations
• Neutral Element / Neutral Elements

On the website Tutorela you will find a variety of articles on mathematics.

Exponents and roots should always be performed before multiplication or division; and before addition or subtraction.

When working with combined operations that include parentheses, exponents and roots, multiplications and divisions, as well as additions and subtractions, we must use the order of operations which indicates the sequence in which we should perform the operations.

• Parentheses are solved first. If there are parentheses within other parentheses, we solve the inner ones first and then the outer ones.
• Exponents and roots are solved next.
• Multiplications and divisions are solved (from left to right).
• Additions and subtractions are solved (from left to right).

Powers are used to abbreviate the multiplication of a number (called the base) by itself, a certain number “n” of times.

When we have combined basic operations that include powers, we must remember that powers are to be resolved after the parentheses.

We solve the parentheses, and subsequently we raise the bases to the indicated exponent, this is done by multiplying it by itself, as many times as the exponent indicates.