Math 9 – What I have learned about Grade 9 exponents

The concept of exponents

Exponents are used to multiply a number by itself a number of times. This is a number with an exponent.

1(3)^2

The 3 is the base of the number. It can be a whole number, negative number, decimal number or even a fraction.

The 2 is the exponent. The exponent dictates how many times to multiply the base by itself, and like the base, can be a whole number, negative number, decimal number or a fraction.

The 1 is the coefficient. When no coefficient is specified, it defaults to 1. The coefficient is multiplied by the base to the power of the exponent. If the exponent is 0, the coefficient instead takes the place of the base.

Exponents are simply an easy way to display large numbers/expressions. This is the same number, but in expanded form, which is to say, the expression which the exponent specifies.

1 \cdot 3 \cdot 3

As you can see, 1(3)^2 is simply 1 \cdot 3 \cdot 3 when expanded.

Here’s an example of exponent numbers with negative bases.

-2^3 (-2)^3

Upon first glance, it may seem as if the two numbers are identical, but no. Exponents will only fully copy numbers within brackets. An example:

-2^3 = -2 \cdot 2 \cdot 2

(-2)^3 = (-2) \cdot (-2) \cdot (-2)

In the first expression, only the first 2 was negative, while all the 2s in the second expression are. This is because of the brackets.

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The difference between evaluating and simplifying

Many people are confused when it comes to evaluating and simplifying.

Simplifying is the reduction of an expression into its most basic form. When simplifying, do not multiply the coefficient.

ie: 4^4\cdot 4^4 = 4^8

Evaluating is the operation of an expression.

ie: 4^4\cdot 4^4 = 65536

 

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Multiplication law

When multiplying two numbers with identical bases that have exponents, it is possible to simplify them.

2^4\cdot 2^6

To simplify, we add the exponents together.

4 + 6 = 10

As we are just simplifying, we don’t need to multiply the base number, so we keep the same base.

2^10

If the base is negative, the answer stays the same, but with a negative base in the final simplification. An example:

-2^4\cdot 2^6

4 + 6 = 10

-2^10

If the exponents are negative, it works the same way it does with regular numbers, with adding negative numbers actually subtracting from positive numbers for example.

2^-4 \cdot 2^6

-4 + 6 = 2

-2^2

2^-4 \cdot 2^-6

-4 + (-6) = -10

2^-10

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Division law

Dividing two numbers with identical bases and exponents is similar to the multiplication law, except instead of adding, you subtract.

3^8\div  3^4

8 – 4 = 4

3^4

A negative base works the same way as multiplications.

-3^8\div  3^4

8 – 4 = 4

-3^4

And the same with negative exponents, with the exponents instead being added together.

3^-8 \div  3^4

-8 – 4 = 12

3^12

3^-8 \div  3^-4

-8 – (-4) = -4

3^-4

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Power of a power law

A power of a power is the term given to a number with an exponent in brackets that have an exponent, like so.

(2^3)^3

With an expression like this, it is possible to simplify it to a regular number with an exponent. In this case, we would multiply the exponents.

3 \cdot 3 = 9

2^9

As with the other laws, multiplying negative exponents is the same as multiplying normal negative numbers.

(2^{-3})^3

-3 \cdot 3 = -9

2^{-9} (2^{-3})^{-3}

-3 \cdot -3 = 9

2^9

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Exponents on variables

This is an example of an expression with exponents and variables.

4x^{2}y^{3}\cdot 4x^{3}y^{2}

When simplifying this, you will want to add together the exponents of each variable separately, like so.

2 + 3 = 5

3 + 2 = 5

4x^{5}y^{5}

When simplifying an expression with division, you will instead want to subtract.

6x^{5}y^{4}\div 6x^{2}y^{7}

5 – 2 = 3

4 – 7 = -3

6x^{3}y^{-3}

Evaluating expressions with exponents and variables is not much different to simplifying them, as values still need to be assigned to the variables to get a result, therefore, you cannot evaluate further once you reach a single number.

3x^{3}y^{4}\cdot 4x^{6}y^{2}

You will want to multiply/divide the base number as usual.

3 \cdot 4 = 12

3 + 6 = 9

4 + 2 = 6

12x^{9}y^{6}

3x^{3}y^{4}\div 4x^{6}y^{2}

3 \cdot 4 = -1

3 + 6 = -3

4 + 2 = 2

-1x^{-3}y^{2}

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