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Integer Exponents
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 About negative exponents!
 How to do operations with negative exponents.
 That we can use this knowledge to help us look at microorganisms, help wildlife, and even learn about elements!

Discussion Questions

Before VideoWhat is an exponent? What are the names of the different parts of an exponent?
ANSWER
Exponents represent repeated multiplication. Exponents have a base and a power.

4 is the base (the number being multiplied) and 9 is the power (the number of times 4 is multiplied).

49 = 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 262,144

When I multiply two exponents with the same base, I can keep the same base and add the powers. Sample answer: 2^{3} × 2^{2} = 2^{3} + 2 = 2^{5}.

When I divide two exponents with the same base, I can keep the same base and subtract the powers. Sample answer: 2^{5} ÷ 2^{3} = 2^{5 – 3}= 2^{2}.


After VideoWhat is the relationship between the following expressions: 2^{3}, 2^{2}, 2^{1}, 2^{0}, 2^{–1}, 2^{2}ANSWER

Each exponential expression is 12 the value of the one before it: 8, 4, 2, 1, [ggfrac]1/2[/ggfrac], [ggfrac]1/4[/ggfrac].

A number raised to a negative power is the same as 1 over that same number raised to the positive power.

2 ×[ggfrac]1/10^{23}[/ggfrac] or [ggfrac]2/10^{23}[/ggfrac]

The rule doesn’t change. Keep the same base and add the powers. The only change is that I may be adding a negative number.

The rule doesn’t change. Keep the same base and subtract the powers. The only change is that I may be subtracting a negative number.



Vocabulary

Exponential expression
DEFINE
An exponential expression is another way of showing repeated multiplication. 2^{3} = 2 × 2 × 2.

Base
DEFINE
In an exponential expression, the base represents the number that is being multiplied. In 2^{3}, the 2 is the base.

Power
DEFINE
In an exponential expression, the power represents the number of times the base is multiplied by itself. In 2^{3}, the 3 is the power.

Product of powers rule
DEFINE
A rule of arithmetic with exponential expressions. It says that when you multiply two exponential expressions with the same base, you can keep the base and add the exponents.

Quotient of powers rule
DEFINE
A rule of arithmetic with exponential expressions. It says that when you divide two exponential expressions with the same base, you can keep the base and subtract the exponents.

Equivalent expression
DEFINE
Two expressions are equivalent if they represent the same quantity. For example, 2^{3}and 2 × 2 × 2 are equivalent expressions because they are both equal to 8.

Exponential expression
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageWHAT IS AN EQUIVALENT EXPRESSION?Equivalent expressions are expressions have the same value. You can use the rules of integer exponents to write expressions with exponents in simpler ways.
To better understand integer exponents…
WHAT IS AN EQUIVALENT EXPRESSION?. Equivalent expressions are expressions have the same value. You can use the rules of integer exponents to write expressions with exponents in simpler ways. To better understand integer exponents…LET’S BREAK IT DOWN!
Exponents represent repeated multiplication.
An exponential expression has a base and a power. In 2^{3}, 2 is the base and 3 is the power. The base tells you the number that is multiplied. The power tells you how many times to multiply the base. So, 2^{3} = 2 × 2 × 2. Similarly, 3^{4} = 3 × 3 × 3 × 3. Now you try: Rewrite 2^{5}as a multiplication expression.
Exponents represent repeated multiplication. An exponential expression has a base and a power. In 23, 2 is the base and 3 is the power. The base tells you the number that is multiplied. The power tells you how many times to multiply the base. So, 23 = 2 × 2 × 2. Similarly, 34 = 3 × 3 × 3 × 3. Now you try: Rewrite 25as a multiplication expression.Negative exponents represent fractions.
Consider the following equations. 10^{3} = 10 × 10 × 10. 10^{2} = 10 × 10. 10^{2} is ten times less than 10^{3}. 10^{1} = 10. 10^{1} is ten times less than 10^{2}. 10^{0} = 1, which is ten times less than 10^{1}. To continue this pattern, 10^{–1} must be ten times less than 1, which is [ggfrac]1/10[/ggfrac]. 10^{–2} =[ggfrac] 1/10 × 10[/ggfrac], which is [ggfrac]1/100[/ggfrac]. 10^{–3} = [ggfrac]1/10× 10× 10[/ggfrac], which is [ggfrac]1/1,000[/ggfrac]. An exponent with a negative power represents 1 over the same exponent with a positive power. 10^{–2 = [ggfrac]1/102[/ggfrac]. This works for all negative exponents, such as 2–6 = [ggfrac]1/26[/ggfrac]. Now you try: Rewrite 3–4 as an exponent with a positive power.}
Negative exponents represent fractions. Consider the following equations. 103 = 10 × 10 × 10. 102 = 10 × 10. 102 is ten times less than 103. 101 = 10. 101 is ten times less than 102. 100 = 1, which is ten times less than 101. To continue this pattern, 10–1 must be ten times less than 1, which is [ggfrac]1/10[/ggfrac]. 10–2 =[ggfrac] 1/10 × 10[/ggfrac], which is [ggfrac]1/100[/ggfrac]. 10–3 = [ggfrac]1/10× 10× 10[/ggfrac], which is [ggfrac]1/1,000[/ggfrac]. An exponent with a negative power represents 1 over the same exponent with a positive power. 10–2 = [ggfrac]1/102[/ggfrac]. This works for all negative exponents, such as 2–6 = [ggfrac]1/26[/ggfrac]. Now you try: Rewrite 3–4 as an exponent with a positive power.Use negative exponents to find the length of an amoeba.
An amoeba has length 7 × 10^{–6} m. First, evaluate the exponent: 10^{–6} = [ggfrac]1/10^{6}[/ggfrac] = [ggfrac]1/1,000,000[/ggfrac]. That's one millionth. 7 × one millionth is 7 millionths. The amoeba is 7 millionths of a meter long.
Use negative exponents to find the length of an amoeba. An amoeba has length 7 × 10–6 m. First, evaluate the exponent: 10–6 = [ggfrac]1/106[/ggfrac] = [ggfrac]1/1,000,000[/ggfrac]. That's one millionth. 7 × one millionth is 7 millionths. The amoeba is 7 millionths of a meter long.Multiply and divide exponential expressions with integer exponents.
Multiply and divide exponential expressions with integer exponents. Previously, you learned that when you multiply exponential expressions with the same base, you can add the powers. For example, 23 × 25 = 23+5 = 28. The same rule applies when any of the powers are negative. 43× 48 = 43+8 = 45. When you divide exponential expressions with the same base, you can subtract the powers. This rule also works with positive and negative powers. For example, [ggfrac]74/72[/ggfrac]=742=76.Many careers use negative exponents.
Here are three examples from many careers that use negative exponents. Microbiologists use negative exponents to represent the size of very small organisms, like bacteria. Veterinarians can represent the weight of very light animals using negative exponents. Chemists use negative exponents to represent the weights of elements and compounds.
Many careers use negative exponents. Here are three examples from many careers that use negative exponents. Microbiologists use negative exponents to represent the size of very small organisms, like bacteria. Veterinarians can represent the weight of very light animals using negative exponents. Chemists use negative exponents to represent the weights of elements and compounds. 
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Science & Math$_{/yr}

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Rewrite 3^{6} as a single rational number.
Rewrite 3^{6} × 3^{4} as a single rational number.
Rewrite 3^{6} ÷ 3^{4} as a single rational number.
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