We Will Discuss Exponent vs Power.
Powers and exponents are tools to rewrite long multiplication problems easily in mathematics, especially in algebra. Algebra is known to be one of the key branches of mathematics which deals primarily with the concept of number theory. It can also be referred to as the study of mathematical symbols. You might have noticed superscripts in mathematical relationships; superscripts can be defined as the one that is placed above to the right of a number. This is known as an exponent and the whole expression can be called as exponentiation.
The operation involves two numbers that can be written in this way xa, where ‘x’ is equal to the base number, and ‘a’ can be defined as the exponent. The exponent can be basically known as a superscript that is used to simplify larger mathematical problems. The whole expression is known as “power” and written as “x to the power of a” where ‘a’ is any positive integer.
What is Power in Mathematics?
Power can be defined as a mathematical expression that can be used to represent exactly how many times a number should be used in a multiplication process. In simple words, it is an expression that describes repeated multiplication of the same given number. Power in mathematics is written as “raising a number to the power of any other number”.
Let’s consider the following example:
3 × 3 × 3 × 3 this is equal to 81. This can also be written in this manner 34 = 81. This is an exponential notation and it simply means the number the number ‘3’ is to be multiplied four times by itself to get the number 81 or in other words, we can say “3 raised to the power of 4” or “3 raised to the 4th power” gives us 81. The number ‘3’ is known as the base number and ‘4’ is known as the power or exponent.
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What is an Exponent in Mathematics?
Let’s discuss what Exponent is generally used interchangeably with power but it is used in a different context. While power is used to represent the whole expression, but the exponent is the superscript placed above to the right of the base of any number. It is generally defined as a positive or negative number which represents the power to which the base number is raised meaning it states the number of times a number needs to be used in a multiplication process.
For example, in \[5^{3} = 5*5*5\] equals to 125, the base number is equal to ‘5’ which is used thrice in a multiplication meaning over here we are multiplying 5 three times by itself. Exponents generally go by powers or indices. Square and cube are the two most commonly used exponents in geometry.
For example, ‘a2’ is defined as ‘a square’ and ‘a3’ is defined as ‘a cube’. If the exponent is equal to 1, then the result is the base number and if the exponent is 0, then the result is always equal to 1. For example, 21 is equal to 2 and 20 equals to 1.
Here are the seven laws of exponents:
Laws of Exponents:
Difference Between Power and Exponent with Example (Exponent vs Power)
Let’s list down the difference between exponent and power with example.
Definition
In mathematical relationships, power is referred to the number of times a number is multiplied by itself meaning the number you get raising a number to an exponent whereas an exponent can be defined as the number of times the number is used in a multiplication.
Exponents are often known as powers or indices. In simple terms, power can be defined as an expression that represents repeated multiplication of the same number whereas exponent is the quantity that represents the power to which the number is raised. Both these terms are often used interchangeably in mathematical operations.
Example of Power and Exponent:
The expression 5 × 5 × 5 can always be written in a shorter way as \[5^{3}\] using the concept of exponents, 5 × 5 × 5 = \[5^{3}\]. The expression basically represents repeated multiplication of the same number which is known as power. Here’ the number ‘5’ represents the base and the number ‘3’ represents the exponent and the whole expression denotes that “5 to the power of 3” or “5 to the third power” which means that over here 5 is multiplied by itself three times.
Similarly, \[2^{5} = 2*2*2*2*2\] is equal to 32.The expression basically denotes “2 to the power of 5” or “2 to the 5th power”. In Mathematics, exponents make it easy to write and use multiplications factor.
Questions to be Solved:
Question 1) Find the value of the\[2^{5}\].
Solution) To find the value of \[2^{5}\]
We need to multiply the base that is equal to 2, and the exponent tells us how many times the number needs to be multiplied by itself.
\[2^{5} = 2*2*2*2*2\] equals 32.
When a number of times a number is multiplied by itself, it is known as the exponent of that number. For example 3 to the 2nd which can also be written as \[3^{2}\] means:
3*3=9
\[3^{2}\] is not the same as 3*2=6.
Any number that is raised to power 1 will be the number itself. For example,
\[x^{1}=x\]
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Important Rules Related to Exponents
\[x^{0}=1\]
When the exponent is 0, the expression will always be equal to 1.
150=1
123750=1\[x^{-1} = \frac{1}{x^{m}}\]
When an exponent is a negative number, the result will always be a fraction. Fractions are represented in the forms of numerators over denominators. In this case, the value of the numerator will always be equal to 1.
\[6^{-2} = \frac{1}{6^{2}}\]
\[2^{-4} = \frac{1}{2^{4}}\]
What are Powers?
The expressions representing the repeated multiplication of the same factor are known as powers. For example, 44 can be written as 42 where 4 is the base and 2 is the exponent.
5 to the second power is 52 and is also known as 5 square.
Important Rules Related to Powers
Powers can be multiplied if the bases of two powers are the same. While multiplying two powers, their exponents are added. For example-
\[a^{x} * a^{y} = a^{x+y}\]
\[2^{2} * 2^{1} = 2^{2+1}\]
\[= 2^{3}\]
Powers can be divided if the bases of the two powers are the same. While dividing two powers, their exponents are subtracted. For example-
\[\frac{a^{x}}{a^{y}} = a^{x-y}\]
\[\frac{a^{2}}{2^{1}} = 2^{2-1}\]
\[ = 2^{1}\]
When a product is raised to a power, then each of the factors is raised with power.
\[\left ( a.y\right )^{x} = a^{x}.y^{x}\]
\[\left ( 3.y\right )^{x} = 3^{2}.y^{1}\]
\[= 9y^{2}\]
Combining the power of a product and the power of a power rule, we get
\[\left ( a^{x}.b^{y} \right )^{z} = a^{xz}.b^{yz}\]
\[\left ( a^{2}.b^{2} \right )^{3} = a^{2*3}.b^{2*3}\]
\[= a^{6}.b^{6}\]
Difference Between Exponents and Powers
FAQs on Difference Between Power and Exponent for JEE Main 2024
1. Why are powers and exponents important?
In mathematics, to represent repeated multiplications, powers and exponents are used therefore they are a very important part of mathematics. Power and exponents are used in different contexts, yet power is synonymous with exponents. Sometimes when there are a large number of times a multiplication is to be done, powers and exponents seem to be very useful as they allow us to abbreviate and make the calculations easier. Suppose if we want to write 4 is multiplied 10 times with itself, then instead of writing 4 x 4 10 times, we can simply write 410.
2. What are the uses of powers and exponents in real life?
Powers and exponents are proven to be very useful in some scientific applications such as the Richter scale or the pH scale while taking measurements and while writing very small and very large numbers. Exponents are also used in physics engines of computer games, in economics while showing the exponential growths in debts, while measuring distance, light and sound, and many more. Biologists, economists, bankers, physicists, mathematicians, geologists, statisticians, and many other professionals use exponents in their daily lives.
3. What are the most important laws of exponents?
The most important laws of exponents are as follows-
Product of Power Law- When the power of the same bases are multiplied, the base remains the same and the exponents get added.
The Quotient of Power Law- When the power of the same bases are divided, the base remains the same and the exponents get subtracted.
Power of a Power Law- In this law, power is raised to a power.
Power of Product Law- When a base is multiplied by an exponent, the exponent is distributed to each base.
Power of Quotient Law- In this law, power is raised to a quotient.
Zero Power Law- Any base with power 0 equals 1.
Negative Exponent Law- When an exponent is negative, the reciprocal of it turns it into a positive.
4. How are exponents represented while writing?
Exponents are the terms representing the number of times one number is multiplied by itself. Exponents can be written in the form of numeric values as well as variables. Just like we can have a variable to a given power, we can also number to power with a variable. For example- 2x which means 2 would be multiplied by itself x times, where x can be any value.
5. Simplify the expression- \[\left ( x^{4}.y^{2}.z^{1} \right )^{2}\].
\[ = x^{4*2}.y^{2*2}.z^{1*2} \]
\[ = x^{8}.y^{4}.z^{2} \]
6. What is an Exponent and What is Power in Math?
Let’s know about exponent and power. An expression that represents repeated multiplication of the same factor is known as a power. For example, 52 here, the number 5 is known as the base, and the number 2 is known as the exponent. The exponent is 2 that corresponds to the number of times the base is used as a factor.
7. What is Power in Math?
The power (or the exponent) of a number basically defines how many times to use the number in the multiplication process. It is written as a small number to the right and above the base number whose power we need to find.
8. What are the 7 Laws of Exponents?
Here are the seven laws of exponents:
Multiplying the powers that have the same Base.
Dividing Powers having the same Base.
Power of a Power.
Multiplying Powers that have the same Exponents.
The concept of Negative Exponents.
Power with Exponent equal to Zero.
Fractional Exponent.