Exponents and Powers: An Overview
When the numbers are really huge, such as 54,32,00,00,000, reading them is difficult, thus we express them as exponents. Exponents make it simple to read, write, interpret, and compare these numbers.
The repeated multiplication of the same integer with respect to the number of times is expressed by exponents, for example, 2^{3} = 2 \( \times \) 2 \( \times \) 2 = 2 \( \times \) 2 \( \times \) 2 = 2 \( \times \) 2 \( \times \) 2 = 2 \( \times \) 2. Exponents are used to raise a number to a power, with the exponent being the power. The base of the power is the number to which the power is raised, so in 23, 2 is the base and 3 is the exponent. The distinction between power and exponents is that the exponent is the power multiplied by the base number, whereas power is the sum of the base number and the exponent. As a result, the number 23 denotes strength.
Exponents are used to write huge numbers in a concise manner.
Here, 8 is the base, 3 is the exponent and 83 is the exponential form of 512.
This can be read as “8 raised to the power of 3.”
Definition of Exponent
The exponent tells us how many times a number should be multiplied by itself to get the desired result. Thus any number (for example, ‘a’) raised to power ‘n’ can be expressed as:
a^{n} = a \( \times \) a \( \times \) a \( \times \) a \( \times \) a \( \times \) a…. \( \times \) a (n times). Here, a can be any number and n is the natural number.
an is also called the n^{th} power of a. In this, ‘a’ is the base and ‘n’ is the exponent or index or power. ‘a’ is multiplied ‘n’ times - it is a method of repeated multiplication.
Note:
When we write a natural number in expanded form, we can also write it in exponential form.
For example:
247983 = 2 × 100000 + 4 \( \times \) 10000 + 7 \( \times \) 1000 + 9 \( \times \) 100 + 8 \( \times \) 10 + 3 \( \times \) 1
= 2 × 105 + 4 \( \times \) 104 + 7 \( \times \) 103 + 9 × 102 + 8 \( \times \) 101 + 3 \( \times \) 1
Some Important Points to Remember
Students must master seven exponent rules, sometimes known as exponent laws. Each rule shows how to solve several types of arithmetic equations as well as multiply, divide, and add exponents.
Make sure you go over each exponent rule in class because they all play a part in solving exponent-based equations.
1. The rule of the product of powers
When multiplying two bases of the same value, keep the bases the same and add the exponents together to get the result.
4^2 \( \times \) 4^5 =?
Because the base values are both four, keep them the same and then multiply by the exponents. (2 + 5) together.
4² \( \times \) 4^5 = 47
To get the answer, multiply four by itself seven times.
47 = 4 \( \times \) 4 \( \times \) 4 \( \times \) 4 \( \times \) 4 \( \times \) 4 \( \times \) 4 = 16,384
Adding the exponents together in an equation like this is a quick way to get the answer.
Here's a more difficult question to consider:
(4x^2)(2x^3) =?
Because the coefficients do not have the same base, multiply them together (four and two). Then add the exponents while keeping the 'x' the same.
(4x^2)(2x^3) = 8x^5
2. The rule of the power quotient
The quotient rule is the polar opposite of the product rule, just as multiplication and division are diametrically opposed. When dividing two bases with the same value, keep the base fixed and subtract the exponent values.
5^5 + 5^3 =?
Both bases in this equation remain the same because they are both five. Then, using the exponents, subtract the divisor from the dividend.
5^5 + 5^3 = 5^2
Finally, simplify the equation if necessary:
5² = 5 \( \times \) 5 = 25
3. A power rule's authority
This rule describes how to solve problems involving the boosting of one power by another.
(𝒙^3)^3 =?
In equations like the one above, multiply the exponents together while keeping the base constant.
(𝒙^3)^3 = 𝒙^9
4. The rule of product power
When multiplying any base by an exponent, distribute the exponent to each component of the base.
(xy)3 =?
In this equation, the power of three must be distributed to both the x and y variables.
(xy)3 = x3y3
This rule also applies if the base has exponents attached to it.
x6y6 = (x2y2)3
In this equation, both of the variables are squared and raised to the power of three. That is, the exponents of both variables are multiplied by three, resulting in variables raised to the power of six.
5. The quotient rule's power
A quotient is just the result of dividing two values. In this rule, you are raising a quotient by a power. Like the power of a product rule, the exponent must be spread within all values in the brackets to which it is associated.
(x/y)4 = ?
Multiply both variables within the brackets by four in this case.
6. The rule of zero power
Any base raised to the power of zero is equal to one.
The simplest way to describe this concept is to use the quotient of powers rule.
4^3/4^3 =?
Using the quotient of powers method, subtract the exponents from each other, which cancels them out and leaves only the base. When a number is divided in half, it equals one.
4^3/4^3 = 4/4 = 1
Whatever is raised to the power of zero produces one, no matter how long the equation is.
7. The rule of the negative exponent
When a negative exponent is used to raise a number, convert it to a reciprocal to make the exponent positive. Don't use the negative exponent to make the base negative.
Consider the following exponent example:
x^(-2) =?
Use the following formula to reciprocate a number:
Using the Standard Form to Express Large Numbers
Exponents are a useful way to express large numbers. If a number can be stated as k 10m, where 1 is the first digit, k 10 is the tenth digit, and m is a natural number, it is said to be in standard form.
The exponent of 10 in standard form is one less than the digit count (number of digits) to the left of the decimal point of a given number.
e.g. 10,000 = 10 \( \times \) 10 \( \times \) 10 \( \times \) 10 = 10^4.
10 is the base and 4 is the exponent.
85 = 8.5 \( \times \) 10 = 8.5 \( \times \) 10^1
850 = 8.5 \( \times \) 100 = 8.5 × 10^2
8500 = 8.5 \( \times \) 1000 = 8.5 × 10^3
8500 = 8.5 \( \times \) 10000 = 8.5 × 10^4
and so on.
Some important points:
100=10^2 (It can be read as 10 raised to 2)
512=8^3
243=3^5
Here, 10, 8 and 3 are the bases, whereas 2, 3 and 5 are their respective exponents.
We also can say:
100 is the 2nd power of 10,
512 is the 3rd power of 8,
243 is the 5th power of 3, etc.
Numbers in exponential form satisfy the following laws for all non-zero integers a and b, as well as whole numbers m and n:
a. a^m × a^n = a^(m+n)
b. a^m ÷ a^n = a^(m-n) , m n
c. (a^m)^n = a^(mn)
d. a^m × b^m = (ab)^m
e. a^m ÷ b^m = (a/b)^m
f. a^0 = 1
g. (-1)^(even number) = 1
h. (-1)^(odd number) = -1
In algebra, exponents are a helpful tool. We can rewrite repeated multiplication with exponents. The exponent is the number of times a number is multiplied by itself. Any number multiplied by 0 equals 1. When rewriting an expression with exponents, however, we must pay attention to the placement of negatives and parentheses. Positive, negative, zero, and rational/fractional exponents are the four types of exponents. Download the Class 7 Maths Chapter 13 extra questions PDF from the MSVGo website or app to learn more about this chapter.
Website: https://msvgo.com/
1. What is standard form, and how does it differ from other forms?
Answer. Any number between 1.0 and 10.0 is frequently stated as a decimal number, including 1.0 multiplied by a factor of 10. This type of variant is referred to as its standard form.
2. What is the difference between powers and exponents?
Answer. Exponents are sometimes known as powers or indices. In other words, power denotes an expression that reflects repeated multiplication of the same number, whereas exponent denotes the power to which the number is raised. In most mathematical operations, we use both of these names interchangeably.
3. As explained in Chapter 13 of Maths in Class 7, what do you mean by exponents?
Answer. Exponents are a type of expression that allows you to write a huge number in a concise and complex way that is easy to read, understand, compare, and work with. For example, reading 10,00,000 in this form is tough, but we can write it as 106 and read it as 10 raised to the power of six with the use of exponents. The lengthy numeral is simple to read and grasp when exponents are used.