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Like numbers, powers can also be negative too. Solving negative power has a small twist and that is why in this lesson you will learn how to solve negative exponents. What is power? It is the expression that shows repeated multiplication of the same number.  For example, { 2 }^{ 4 } means 2 \times 2 \times 2 \times 2 which is equal to 16. Hence, we can write { 2 }^{ 4 } = 16. Same rules are applied to any number with an exponent but what if the number is a negative number? For example, { (-2) }^{ 4 }, to solve this, you need to break it into two steps. The first step is to calculate the absolute value which is 2 \times 2 \times 2 \times 2 = 16. Once you find the absolute value, now proceed to the sign multiplication. If the exponent is an even number then the resultant sign will always be positive but if the exponent is an odd number then the sign will always be negative. In the above example, the exponent is even hence the answer will be +16. Let's do another example for your clarification, for example, { (-2) }^{ 3 }, first, calculate the absolute value: 2 \times 2 \times 2 = 8. Now to find the resultant sign, check whether the exponent is an even number or odd number. The exponent is 3 which is an odd number, which means the resultant sign will be negative so the answer of { (-2) }^{ 3 } is -8.

Let's proceed to negative exponents. Negative exponents mean that the power is in negative form. For example, { a }^{ -n }. If you see a negative sign on the power, that is an indication that you are dealing with a negative exponent. The question is how to solve the negative exponent? You can't perform multiplication of factors because we use to do that with positive exponents and that is why we will be changing the negative exponents to positive exponents. To change the negative exponent to a positive exponent, you need to reciprocate the number. So, here you have a negative exponent, { a }^{ -n }, reciprocate it to make it a positive exponent.

{ a }^{ -n } = \frac { 1 }{ { a }^{ n } }

Since the negative sign is removed, now you can multiply the number by itself as you use to do before. However, do note this that the value of a can't be equal to 0, a \neq 0. If the value of a becomes zero that means \frac { 1 }{ 0 } and \frac { 1 }{ 0 } is equal to infinity.

Examples

1. { 5 }^{ -2 }

{ 5 }^{ -2 } = \frac { 1 }{ { 5 }^{ 2 } } = \frac { 1 }{ 25 }

 

2. { 3 }^{ 2 } \times { 3 }^{ -4 }

{ 3 }^{ 2 - 4 } = { 3 }^{ -2 }

{ 3 }^{ -2 } = \frac { 1 }{ { 3 }^{ 2 } }

\frac { 1 }{ { 3 }^{ 2 } } = \frac { 1 }{ 9 }

 

3. { 3 }^{ -2 } \times { 3 }^{ 2 }

{ 3 }^{ -2+2 } = { 3 }^{ 0 }

{ 3 }^{ 0 } = 1

 

Fractional Exponent,   Exponents Worksheets.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.