What is Meant by “Power” in Mathematics?
In mathematics, the term “power” refers to the process of increasing a base number to its exponent. “Base number” and “exponent” are the two fundamental elements of power. The exponent represents the number of times the base number is multiplied, and the base is defined as a number that is multiplied by itself.
What is an Exponent?
The exponent of a number shows the number of times we multiply the number itself. For example, \(2^3\) implies that we multiply 2 by itself three times, which is 2 × 2 × 2.
An exponent is also known as a numerical power. It could be represented as a whole number, a fraction, a negative number, or a decimal.
Visualizing Exponents and Powers?
Number powers can be easily visualized by using shapes and figures. Take a look at the diagram below for more information on the method of visualizing exponents.
Properties of exponents :
The properties of exponents make multiplication and division easier to understand and solve. With this in mind, we will discuss three such properties.
1) The product of powers property
2) The power of a power property
3) The power of a product property
The Product of Powers Property
Add the exponents of powers with the same base while multiplying exponents with the same base number.
\(x^a\times x^b=x^{a+b}\)
Example: \(3^2\times 3^3 = 3^{2 + 3} = 3^5\)
The Power of a Power Property
Multiply the exponents to get the power of a power.
\((x^a)^b = x^{ab}\)
Example: \((5^4)^2 = 5^{4\times 2} = 5^8\)
The Power of a Product Property
To find the power of a product, multiply the power of each factor.
\((xy)^a = x^ay^a\)
Example: \((5\times 8)^5 = 5^5\times 8^5\)
Solved Examples of Product of Powers:
Example 1:
One gigabyte (GB) of computer storage space is \(2^{30}\) bytes. The storage details of a computer are shown below. What is the total storage capacity of the computer?
Solution:
The computer has a total storage capacity of 64 gigabytes. Notice that you can write 64 in the form of a power, \(2^6\).
To solve the problem, use a verbal model.
Total number of bytes
= Number of bytes in a gigabyte \( \times \) Number of gigabytes
\( = 2^{30}\times 2^6 \)
\( = 2^{30 + 6} \)
\( = 2^{36} \)
Hence, the computer has \(2^{36}\) bytes of total storage space.
Example 2:
The lowest altitude of an altocumulus cloud is about \(3^8\) feet. An altocumulus cloud’s highest altitude is roughly three times its lowest altitude. What is an altocumulus cloud’s highest altitude? Write your answer as a power of a number.
Solution:
Given that the lowest altitude of an altocumulus cloud is about \(3^8\) feet, and the highest altitude of an altocumulus cloud is about 3 times the lowest altitude.
So, the highest altitude of an altocumulus cloud \( = 3 \times 3^8\) feet.
\( = 3^1\times 3^8\)
\( = 3^9\) feet.
Example 3:
Every second, the United States Postal Service delivers approximately \(2^4\times 3\times 5^3\) mails. In six days, there are \(2^8\times 3^4\times 5^2\) seconds. How many mails does the United States Postal Service deliver in six days? Make a three-power expression out of your response.
Solution:
Given that the United States Postal Service delivers about \(2^4\times 3\times 5^3\) pieces of mail each second, and there are \(2^8\times 3^4\times 5^2\) seconds in 6 days.
So, the number of mails that would be delivered by the United States Postal Service in 6 days = \(2^4\times 3\times 5^3\times 2^8\times 3^4\times 5^2\)
\( = 2^{4 + 8}\times 3^{1 + 4}\times 5^{3 + 2}\)
\( = 2^{12}\times 3^5\times 5^5\)
Example 4:
Multiply \(2^4\times 2^2\)
Solution:
The base is the same, which is 2, and we will add the powers according to the rule.
\(2^4\times 2^2\)
\( = 2^{(4+2)}\) According to the product of power property
\( = 2^6\)
\( = 64\)
Yes, we can multiply expressions with different coefficients. As shown in the example, the coefficients are multiplied separately.
\(5x^2\times 2x^5\)
\( = (5\times 2)\times (x^2\times x^5)\) According to the product of powers property
\( = 12x^7\)
The powers are added when exponents with the same bases are multiplied.
For example,
\( 5^2\times 5^3 \)
\( =5^{(2+3)}\) According to the product of powers property
\( =5^5 \)
The power outside the parentheses is multiplied with every power inside the parentheses when exponents are multiplied with parenthesis.
For example,
\((3x^4y^5)^3\)
\( = 3^3\times x^{(4\times 3)}\times y^{(5\times3)}\) According to the power of powers property
\( = 27x^{12}y^{15}\)