Basic Exponent Laws I
Exponents are a shorter way to show repeated multiplication of the base. In expanded form, the power x^5, would be written as: x\times x \times x \times x \times x
If you don’t see an exponent on a number or variable, the exponent is 1.
Law of Zero Exponents
You’ll simply need to memorize that for any base, if the exponent is 0, the answer is 1: x^0=1
Anything to exponent zero is 1.
Law of Multiplication
Multiply two powers with the same base: x^5 and x^2: (x^5)(x^2)
If you write this in expanded form: (x \times x \times x \times x \times x)(x \times x)
This would be equivalent to writing: (x \times x \times x \times x \times x \times x \times x)=x^7
We can conclude that: (x^5)(x^2)=x^{5+2}=x^7
This will always be true when multiplying powers with the same base. It is called the Law of Multiplication:
When multiplying powers with the same base, keep the base and add the exponents.
Law of Division
Divide two powers with the same base: x^5 and x^2: \frac{x^5}{x^2}
If you write this in expanded form: \frac{x \times x \times x \times x \times x}{x \times x}
Two factors would cancel from the numerator and denominator: \frac{x \times x \times x \times x \times x}{x \times x}=x \times x \times x=x^3
We can conclude that: \frac{x^5}{x^2}=x^{5-2}=x^3
This will always be true when dividing powers with the same base. It is called the Law of Division:
When dividing powers with the same base, keep the base and subtract the exponents.
Law of Powers
Raise a power to an even higher exponent. For example, raise x^5 to the exponent 2: (x^5)^2
If you write this in expanded form: (x \times x \times x \times x \times x)(x \times x \times x \times x \times x)
This would be equivalent to writing: x \times x \times x \times x \times x \times x \times x \times x \times x \times x=x^{10}
We can conclude that: (x^5)^2=x^{5 \times 2}=x^{10}
This will always be true when raising powers to a higher exponent. It is called the Law of Powers:
When raising a power to a higher exponent, keep the base and multiply the exponents.
Practice
Simplify then evaluate: 2^{11} \div 2^8
SOLUTION
Using the Law of Division: 2^{11} \div 2^8=2^{11-8}=2^3
Evaluating: 2^3=8
Simplify: (m^0)^{10}
SOLUTION
Using the Law of Powers: (m^0)^{10}=m^0
We should more properly use the Law of Zero Exponents to write: m^0=1
Find the value of x that makes the equation true: 4^x \times 4^4=4^7
SOLUTION
Since we are multiplying the two powers, we use the Law of Multiplication: x+4=7
Therefore: x=3
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