Introduction to Word Problems Involving Exponents
Infinity is a floorless room without walls or ceilings.
—AUTHOR UNKNOWN
This lesson will explain the rules for exponents and the various forms of numbers. Scientific notation examples and word problems involving exponents will also be demonstrated.
The Basics of Exponents
Exponents are a way of writing large and small numbers in a shortened way. In the expression 2^{3}, the number 2 is the base and the number 3 is the exponent. The exponent tells how many times the base is used as a factor, or how many times it should be multiplied. For example, the expression 2 to the third power is written as 2^{3}= 2 × 2 × 2 = 8. 2^{3}is the exponential form, 2 × 2 × 2 is the expanded form, and 8 is the standard form of this number.
Tip:
There are three useful forms of numbers when working with exponents. They are:

Exponent Rules
The rules of exponents can be explained by using the different forms of numbers.
Multiplying
The expressions 4^{5}and 4^{2}have the same base. In expanded form, 4^{5}= 4 × 4 × 4 × 4 × 4 and 4^{2}= 4 × 4. To multiply them together, the expression becomes 4^{5}× 4^{2}= 4 × 4 × 4 × 4 × 4 × 4 × 4. This expanded form is multiplying seven 4s together; in other words, 47. This pattern can be explained as 4^{5}× 4^{2}= 4^{5 + 2}= 4^{7}. When multiplying with like bases, add the exponents.
Dividing
To divide like bases, such as 3^{4}÷ 3^{3}, use expanded form and write division as a fraction to find a pattern. The expression becomes . Notice how the common factors in the numerator and denominator cancel. This pattern can be summarized as 3^{4}÷ 3^{3}= 3^{4 – 3}= 3^{1}= 3. When dividing like bases, subtract the exponents.
Raising a Power to Another Power
When you are raising an exponent to a power, such as (2^{4})^{3}, the base becomes the expression within the parentheses. Therefore, to simplify this expression, rite the base 2^{4}as a factor three times. This is 2^{4}× 2^{4}× 2^{4}= 2^{4 + 4 + 4}= 2^{12}. When raising a power to another power, multiply the exponents.
Tip:
The rules for exponents can be summarized as the following:
 When multiplying like bases, add the exponents.

 x^{3}× x^{4}= x^{3 + 4} = x^{7}
 When dividing like bases, subtract the exponents.
 When raising a power to another power, multiply the exponents.

 (x^{3})^{2}= x^{3 × 2 }= x^{6}
 When adding or subtracting expressions with exponents, follow the order of operations.

Negative Exponents
A negative exponent tells you to take the reciprocal of the base. For example,. For expressions to be simplified, they should be written with positive exponents.
Converting to and From Scientific Notation and Standard Form
Scientific Notation
Scientific notation is used to write very large and very small numbers in a more efficient manner. This form uses a number between one and ten and multiplies it by a power of 10. For example, the number 3,000,000,000 is written as 3 × 10^{9} in scientific notation.
Converting From Standard Form to Scientific Notation
To change a number from standard form into scientific notation, take the first nonzero digit and place a decimal point to its right to form a value between 1 and 10. Then, multiply by a power of 10, where the exponent is the number of places the decimal point moves.
For example, to write the number 4,500,000 in scientific notation, use the digits 4 and 5 and write the decimal 4.5. Because the decimal point needs to move six places to the left to be between the 4 and 5, the exponent is 6. The scientific notation is 4.5 × 10^{6}.
Converting From Scientific Notation to Standard Form
To convert to standard form from scientific notation, reverse the process just explained. Write the value between one and 10, but move the decimal point as according to the exponent.
For example, for the value 3.42 × 10^{–4}, first write the number 3.42. Then, move the decimal to the right if the exponent is positive or to the left if the exponent is negative. Use zeros as placeholders when necessary. Since this exponent is –4, move the decimal four places to the left. The standard form is 0.000342.
Tip:
When you are converting from standard form to scientific notation, numbers greater than one have a positive exponent and numbers less than one have a negative exponent.

Word Problems with Exponents
The word problems in this section involve using the properties of exponents. Use the information in this lesson and the problem solving steps to solve each one.
Example 1
What is the product of 10^{5}× 10^{8}?
Read and understand the question. This question is looking for the product of two values that have the same base.
Make a plan. Add the exponents to find the solution.
Carry out the plan. The problem becomes 10^{5}× 10^{8}= 10^{5 + 8} = 10^{13}.
Check your answer. To check your answer, divide the product by one of the factors that was multiplied in the question.
Because this was the other factor, this answer is checking.
Example 2
What is the quotient of 16^{5}and 16^{3}?
Read and understand the question. This question is looking for the quotient of two values that have the same base.
Make a plan. Subtract the exponents to find the solution.
Carry out the plan. The problem becomes
Check your answer. To check your answer, multiply the quotient by the factor 16^{3}.

 16^{2}× 16^{3}= 16^{2 + 3}
Because this was the first value given in the question, this answer is checking.
Example 3
A garbage company collects 64,000,000 pounds of garbage per year. What is this amount expressed in scientific notation?
Read and understand the question. This question is looking for the scientific notation when the standard form is given.
Make a plan. Use the first nonzero digit and a decimal point to form a number between one and 10, and then write this number multiplied by a power of 10. The exponent is the number of places the decimal point moves from the end of the number to between the 6 and the 4.
Carry out the plan. Write the decimal 6.4. Next, multiply this value by 10^{7}because the decimal moves 7 places to the left. The scientific notation is 6.4 × 10^{7}.
Check your answer. To check your answer, put this number back into standard form. Take the decimal 6.4 and move the decimal 7 places to the right. Add zeros as placeholders where necessary. The number becomes 64,000,000. This answer is checking.
Find practice problems and solutions for these concepts at Word Problems Involving Exponents Practice Questions.