Formula Generation to Find Power
The power of a number demonstrates the repeated multiplication of a number. The number that is multiplied repeatedly is the base, the number of times the base is multiplied to itself is the exponent.
Consider a number a multiplied by itself n times,
a x a x a … n times
A power can be written as:
a x a x a … n times = \(a^n \) ,where a is the base and n is the exponent.
Hence a used as a factor n times is written as \(a^n \).
Powers in words
Let’s understand how to read powers using an example. Consider a power with base 2.
\(2^2\) is read as two squared or two to the second,
\(2^3\) is read as two cubed or two to the third,
\(2^4\) is read as two to the fourth,
\(2^5\) is read as two to the fifth, and so on.
Rapid Recall
Solved Examples
Powers Evaluation Formula
Example 1:
Write the following as Powers. Also, name the base and exponent.
1. 14 x 14 x 14 x 14 … 21 times
Solution:
14 x 14 x 14 x 14 … 21 times = \(14^{21}\)
14 is multiplied by itself 21 times. Hence 14 is the base and 21 is the exponent.
2. 100 . 100 . 100
Solution:
100 . 100 . 100 = \(100^3\)
100 is multiplied by itself 3 times. Hence 100 is the base and 3 is the exponent.
Example 2:
Two squares A and B with their respective side lengths are given below. Find their areas. Which square has a larger area and by how much?
Solution:
Area of a square = side × side = \(side^2\)
To compare two quantities the units of measurement used between both need to be the same. Hence, in square A, let’s convert the side length from inches to feet.
Side of square A = 48 ÷ 12 = 4 feet
Area of square A = \(4^2\)
= 16 square feet
Area of square B = \(3^2\)
= 9 square feet
16 square feet > 9 square feet
Area of square A > Area of square B
Difference in area = 16 – 9 = 7 square feet
Therefore, the area of square A is greater than area of square B by 7 square feet.
Example 3:
Stacey is decorating her house for a party in some time. Her theme for her party is “All Things Paper” so all her decorations are made of paper as well. For one of her decorations, she draws a square on a piece of paper, then she cuts out four smaller squares and one circle from it. How much paper will she have remaining to use for another decoration? (Take \(\pi ~ = ~\frac{22}{7}\))
Solution:
To find the area of the remaining paper, we need to find the sum of areas of the 4 small squares and the area of the circle and subtract that from the area of the larger square.
Area of a square = side × side = \(side^2\)
Area of 1 small square = \(2^2\) = 4 square centimeters
Area of 4 small squares = 4 x 4 = 16 square centimeters
Area of a circle = \(\pi~r^2\)
Area of the circle = \( \frac{22}{7} ~\times~ 7^2\)
= 154 square centimeters
Total area cut out = 16 + 154
= 170 square centimeters
Area of the larger square = \(20^2\)
= 400 square centimeters
Area of the remaining paper = 400 – 170
= 230 square centimeters.
Hence Stacey will have 230 square centimeters of paper remaining.
Example 4:
Travis didn’t pay attention when “powers” as a topic was taught in Math class. He said he had already learnt it. So Travis created the table below and wrote down the respective power values. Are the answers Travis wrote down correct? If not, what are the right answers? Also, can you figure out how Travis got the answers?
Solution:
\(3^2\) = 3 x 3 = 9
\(3^3\) = 3 x 3 x 3 = 27
\(3^4\) = 3 x 3 x 3 x 3 = 81
Hence the values Travis has obtained are incorrect.
To find powers repeated multiplication is used. But Travis has considered repeated addition, that is, multiplication. Each of the values obtained is found by simply multiplying the base by the exponent, which is not the value of power.
3 + 3 = 3 x 2 = 6
3 + 3 + 3 = 3 x 3 = 9
3 + 3 + 3 + 3 = 3 x 4 = 12
The repeated multiplication of the same number can be expressed using power. For instance, 10 to the fourth is \(10^4\) = 10 x 10 x 10 x 10 = 10000 So the value of the power \(10^4\) is obtained by multiplying 10, 4 times.
The number that is multiplied repeatedly is the base and the number of times the number is multiplied is the exponent.
In simple words, power is repeated multiplication. Let’s understand the power of fractions using an example. \((\frac{3}{4})^~=~\frac{3}{4}~\times~\frac{3}{4}~=~\frac{9}{16}\) . Hence to find the power, the product of the numerators and denominators are found, and the answers are expressed as a fraction.
When a negative number is raised to an even number exponent the product becomes a positive number. For instance, – \(8^2\) = -8 x – 8 = 64
The power of a number whose exponent is 1 denotes the number itself. For example, \(7^1\) = 7