Unit 1 Challenge 2 PDF

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Summary

challenge q and a notes...

Description

Select the expression that is correctly evaluated.

Product Property of Exponents

Quotient Property of Exponents

Power of a Power Property of Exponents

Use the properties of exponents to simplify the expression:

For this expression, first use Power of a Power Property, which states that when an exponent is raised to another exponent, the exponents are just multiplied

together. So . Finally, the Quotient Property is used, which states that if exponential expressions with the same base are divided, the

exponents can be subtracted. So

.

Zero Property of Exponents

Any number or expression raised to the zero power will always be 1. Using the quotient rule, subtract exponents

Our Solution, but we will also solve this problem another way Rewrite exponents as repeated multiplication

Reduce three

out of top and bottom

Simplify to exponents

Our Solution, putting these solutions together gives:

Our Final Solution

Negative exponents yield the reciprocal of the base. Properties of Negative Exponents

As a general rule if we think of our expression as a fraction, negative exponents in the numerator must be moved to the denominator, likewise, negative exponents in the denominator need to be moved to the numerator. When the base with exponent moves, the exponent is now positive. This is illustrated in the following example. Negative exponents on b, d, and e need to flip

Our Solution

keep the negative exponents until the end of the problem and then move them around to their correct location (numerator or denominator). As we do this it is important to be very careful of rules for adding, subtracting, and multiplying with negatives. This is illustrated in the following examples: Simplify numerator with product rule, adding exponents

Use Quotient rule to subtract exponents, be careful with the negatives!

Our Solution Simplify the expression to a form in which 2 is raised to a single integer power.

First, use the Power of a Power Property to multiply the exponents 10 and 3. Next, use the Product Property to add the exponents 30 and -10 in the numerator. Finally, use the Quotient Property to subtract the exponents 20 and -7 (be careful of negatives!).

Property of Fractional Exponents

The denominator of a rational exponent becomes the index on our radical. Likewise, the index on the radical becomes the denominator of the exponent. We can use this property to change any radical expression into an exponential expression.

Index is denominator Negative exponents from reciprocals

We can also change any rational exponent into a radical expression by using the denominator as the index.

Index is denominator Negative exponent means reciprocals

The ability to change between exponential expressions and radical expressions allows us to evaluate problems we had no means of evaluating before. We do this by changing the expression to a radical.

Change to radical, denominator is index, negative means reciprocal

Properties of Exponents

Simplify the expression to a single power of x.

Since

is being divided by

, the Quotient Property is used, which

states that the exponents and can be subtracted. However, when adding and subtracting fractions, the denominators need to be the same so change the fraction

to

. Finally, the Power of a Power Product is

applied, which means that the exponents together.

Scientific Notation:

and

can be multiplied

where

The number can be expressed in standard form as ___________. When converting into scientific notation, if we move the decimal to the left, this increases the exponent. If we move the decimal to the right, this decreases the exponent.

HINT

Recall that the decimal number in scientific notation must be at least 1, but no greater than 10. This means that and are not in proper scientific notation. To correct these types of expressions, the decimal needs to shift either to the right or to the left, to fit our rules for what the decimal number can be:



: 0.4 needs to be written as 4.0, and the exponent needs to change from 4 to 3 (decreasing due to a shift to the right).



: 11.2 needs to be written as 1.12, and the exponent needs to change from -2 to -1 (increasing due to a shift to the left).

  A negative exponent means the standard form is a small number. The exponent -3 means that you will move the decimal three places to the left. Correct

Divide the following two numbers in scientific notation.

When two numbers in scientific notation are divided, the front numbers are divided, then the exponent of properties are used to simplify the 10's. 5.6 divided by 1.6 is 3.5; this becomes the first number in the answer. Using the Product Property,

to

divided by

is equivalent

.

o you think we can use any properties of exponents as shortcuts when we raise a number in scientific notation to an exponent power?

Use power rule to deal with numbers and 10's separately

Evaluate Multiply exponents

Our Solution

Generally, numbers written in scientific notation can be expressed as a • 10^n, where a is a decimal number, and 10^n represents a power of ten. In proper scientific notation, there are a couple of restrictions to what “a” is allowed to be. “a” can only contain one non-zero digit to the left of the decimal. Here are some examples of numbers that might look like they are in scientific notation, but they violate this rule for what “a” can be: A zero to the left of the decimal is not allowed More than 1 digit to the left of the decimal is not allowed When moving the decimal in scientific notation, movement to the left increases the power of 10, and movement to the right decreases the power of 10.

HINT

Be careful when increasing and decreasing negative powers of 10. For example, decreasing -3 by 1 makes the number more negative: -4. Similarly, increasing -3 by 1 makes the number less negative: -2.

Add the decimal numbers together, keep the power of 10

Our Solution This tip is worth repeating, especially when adding or subtracting numbers in scientific notation. During the process, you will probably work with numbers that are not written in proper scientific notation. This is okay, because it is required to create a common power of 10. Just be sure that you make final adjustments to your expression, so that the decimal number contains only a single non-zero digit to the left of the decimal. Remember, decimal movement to the left increases the power of 10, and decimal movement to the right decreases the power of 10. Add the following two numbers in scientific notation....