Warm up Use the laws of exponents to simplify the following. Answer should be left in exponential form.

Laws of Exponents

The laws you used in the warm up with integer bases apply to all bases, even variable bases. The laws also apply for all types of exponents, including fractions and decimals. Let’s review the rules and see them applied.

Labeling an exponential expression The expression is written and read as X to the 5th power. X is called the base of the expression. 5 called a power or exponent for the expression. The exponent, 5, tells us that we want X multiplied with its self 5 times Power or exponential form is X5 x  x  x  x  x is Expanded form

Practice Identify the base of Identify the exponent of Is in exponential or expanded form?

ZERO POWER RULE Any base(s) raised to the zero power will always equal 1. Examples

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Product of Powers When two bases are multiplied we add the exponents of the bases. Examples If there are numbers in the expression we can multiple them. Example

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Power of a Power When we have an exponent raised to an exponent we multiple the exponents. Example If there are numbers or more than one variable, inside the parenthesis, they all get raised to the outside power. When we have we first simplify by ‘distributing’ the outside exponent inside, and then, since the two groups are multiplied, we added the exponents of like bases. Examples

Warm up 3-2 Use the laws of exponents to simplify the following. Answer should be left in exponential form.

Quotient of Powers When dividing with exponents we subtraction the exponents of common bases. Examples If there is no other base for you to divide with it is kept in the same place.

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Power of a Quotient When we raising a fraction to a power, we can rewrite the fraction by raising everything on top by the outside exponent and everything on bottom to the outside exponent. Examples

Practice

Negative Exponents If we have a negative exponent we can write it as a positive by taking the reciprocal. If the negative exponent is on the top of a fraction we can write it positive by simply moving it to the bottom of the fraction. If the negative exponent is on the bottom of a fraction we can write it positive by moving it to the top of the fraction. Examples

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Radicals and Exponents Fraction powers can be written as radicals, roots.

Practice