Exponents & The Order of Operations Lesson

Many times, we are faced with a scenario where we have a repeated multiplication of the same number. Suppose we wanted to write the prime factorization of 243. $$243=3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$$ Writing out 5 factors of 3 is quite long and inconvenient. Fortunately, we have exponents that can help to shorten this process. An exponent allows us to write a repeated multiplication of the same number in a more compact form. $$243=3 \cdot 3 \cdot 3 \cdot 3 \cdot 3=3^5$$ Notice how 5 factors of 3 can be written in exponential form as: 3⁵. Where the 3 or larger number is known as the base, and the 5 or smaller number is known as the exponent. The base is the number that is being multiplied by itself in the repeated multiplication. When an exponent is a natural number, it represents the number of factors of the base present in the repeated multiplication. Since 243 can be broken down into 5 factors of 3, we can write 243 as 3 to the 5th power. As another example, suppose we wanted to write the prime factorization of 500. If we didn't use exponents, we could write this out as: $$500=2 \cdot 2 \cdot 5 \cdot 5 \cdot 5$$ Using exponents, we can clean this up quite a bit: $$500=2^2 \cdot 5^3$$ Additionally, we may come across an exponent of 1. When we raise a number to the power of 1, it just gives us back the base. This means that any number raised to the power of 1 is just itself. $$215^1=215$$ $$1397^1=1397$$

Exponents with a Negative Base

When we work with exponents and the base is negative, it can be quite confusing when considering the sign of the answer. Suppose we consider the following: $$-2^2$$ If we punch this up on a calculator, you will get an answer of -4. One might ask why, as we would think that -2² means (-2)(-2), which is clearly 4. So why then does our calculator give us an answer of -4? The reason is simple. The exponent doesn't apply to the negative part unless it is wrapped inside of parentheses. We can actually write -2² as -1 • 2², which makes it clear that the answer is -4. If we want to obtain an answer of +4, we want to make sure to wrap our negative sign inside of parentheses. $$(-2)^2=(-2)(-2)=4$$ $$-2^2=-1 \cdot 2^2=-1 \cdot 4=-4$$ It's important to note that this rule doesn't change the sign if the exponent is odd. This is due to the fact that an odd number of negative factors yields a negative result. $$-3^3=-1 \cdot 3^3=-1 \cdot 27=-27$$ $$(-3)^3=(-3)(-3)(-3)=-27$$

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Order of Operations

The Order of Operations tells us which operation to perform in which order when faced with a problem with multiple operations involved. Most often, people use the acronym PEMDAS to remember the order of operations. We have to be careful when using PEMDAS as following the order of the letters doesn’t always give us the correct answer.
PEMDAS:

1. P » parentheses and other grouping symbols such as absolute value bars, brackets,…etc
   • Always start with the innermost set of parentheses or grouping symbols and work outward. Once inside of grouping symbols, we want to reapply the order of operations.
   • When fraction bars are present, we work above and below any fraction bars separately. We can also wrap the numerator and denominator inside of parentheses and place a "÷" symbol between them to replace the fraction bar.
2. E » exponents and radicals
3. MD » multiply or divide working left to right
4. AS » add or subtract working left to right

The multiply and divide steps, along with the addition and subtraction steps will cause the most confusion. The order in PEMDAS can’t be followed exactly. For example, if we try to follow the sequence of letters with multiplication or division, we may obtain the wrong answer. We can see that the M comes before the D in PEMDAS, but we multiply or divide working from left to right. In the problem:
12 ÷ 3 • 7
We would actually divide first since the division operation is to the left of the multiplication operation. The correct answer is 28:
12 ÷ 3 • 7 = 4 • 7 = 28
As an example, suppose we wanted to find the value for the following expression: $$[10 - (16 \hspace{.1em}÷ \hspace{.1em}(-8))^3] \cdot 7 \hspace{.1em}÷ \hspace{.1em}(-6)$$ First, we want to start with any grouping symbols that are present. In this problem, we have brackets and parentheses. Starting with the innermost set gives us: $$(16 \hspace{.1em}÷ \hspace{.1em}(-8))=(-2)$$ Now, we will replace this in our expression and continue. $$[10 - (-2)^3] \cdot 7 \hspace{.1em}÷ \hspace{.1em}(-6)$$ Now that we have dealt with the innermost set of parentheses, it's time to work outward. Let's next tackle what's inside of the brackets. $$[10 - (-2)^3]$$ Notice how we have a subtraction operation and an exponent operation. The exponent has a higher priority, so we will raise (-2) to the third power first: $$(-2)^3=-8$$ Let's replace this in our expression. $$[10 - (-8)] \cdot 7 \hspace{.1em}÷ \hspace{.1em}(-6)$$ Now, we will perform our subtraction operation inside of the brackets. $$10 - (-8)=10 + 8=18$$ Let's replace this in our expression. $$18 \cdot 7 \hspace{.1em}÷ \hspace{.1em}(-6)$$ Now, we have multiplication and division left. Again, we want to make sure these are worked from left to right. $$18 \cdot 7=126$$ Let's replace this in our expression. $$126 \hspace{.1em}÷ \hspace{.1em}(-6)$$ As our final step, let's perform our division. $$126 \hspace{.1em}÷ \hspace{.1em}(-6)=-21$$ Scroll Right For More »

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