Introduction
A base number being raised to an exponent is referred to as a "power" in mathematics. It signifies that "base number" and "exponent" are the two fundamental components of powers.
Number Power:
How to find the power of a number? How many times to multiply a given number depends on its power (or exponent).
It appears as a small number above and to the right of the basic number.
The small "2" in this illustration tells us to multiply 8 twice:
\[{8^2} = 8 \times 8 = 64\]
Hence the Number Power Formula for 8 times 8 will be \[{8^2}\].
However, the term "power" can also refer to the outcome of an exponent, so in the previous example, "64" is also referred to as the power.
Different Terms of Exponent
Exponent of a Number:
A number's exponent indicates how many times the number has been multiplied by itself. Example: Since 2 is multiplied by itself 4 times, \[2 \times 2 \times 2 \times 2\] can be expressed as \[{2^n}\]. Here, 4 is referred to as the "exponent" or "power," while 2 is referred to as the "base." Generally speaking, \[{x^n}\] denotes that x has been multiplied by itself n times.
Exponent of a Number
Here, Here, in the term xn
x is called the "base"
n is called the "exponent"
And is read as "x to the power of n" or "x raised to n".
Power of Exponents:
We may easily express and represent extremely big and small numbers using exponential notation or the exponential form of numbers. For instance, 10000000000000 is equal to \[1 \times {10^{13}}\] whereas 0.0000000000000007 is equal to\[7 \times {10^{ - 16}}\]. This makes numbers easier to read, helps in ensuring their accuracy, and saves us time.
Properties of Exponents:
These characteristics are regarded as major exponents rules that must be followed when solving exponents. The following list includes exponent qualities.
Product law: \[{a^m} \times {a^n} = {a^{m + n}}\]
Quotient law: \[{a^m} \times {a^n} = {a^{m -n}}\]
Zero Exponent law: \[{a^0} = 1\]
Negative Exponent law: \[{a^{ - m}} = 1 \div {a^m}\]
Power of a Power law: \[{({a^m})^n} = {a^{mn}}\]
Power of a Product law: \[{\left( {ab} \right)^m} = {a^m}{b^m}\]
Important Points for Power:
When a fraction's exponent is negative, we take the fraction's reciprocal to make it positive. Consequently, \[{\left( {\frac{a}{b}} \right)^{ - m}} = {\left( {\frac{b}{a}} \right)^m}\]
We can set the bases to equal when the exponents in an equation are the same on both sides, and vice versa.
Solved Examples:
1: Each tree in a garden has roughly \[{5^7}\] leaves, and there are about \[{5^3}\] trees overall. Calculate the total number of leaves using exponents.
Ans: The number of trees in the garden is \[{5^3}\] , and each tree has \[{5^7}\] leaves. \[{5^3}\]\[ \times \] \[{5^7}\] = \[{5^{10}}\] leaves total, according to the exponents law.
Therefore, there are \[{5^{10}}\] leaves in all.
2: What is 2 when it has a 7 exponent?
Ans: When 2 has an exponent of 7 then the answer will be \[{2^7} = 128\].
As \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\] \[ = 128\].
3: Solve \[{25^3}/{5^3}\].
Ans: Using quotient Law: \[{a^m} \div {a^n} = {a^{\frac{m}{n}}}\]
It is possible to write \[{25^3}/{5^3}\] as \[{(25/5)^3}\]
\[ = {5^3}\]
Conclusion:
An expression known as "power" denotes the process of repeatedly multiplying a value or integer. \[{a^n}\] is often a power where n is the exponent and \[{a^n}\] is the base.
FAQs on Number Power
1. What does 'number power' mean in mathematics?
In mathematics, 'number power' is a shorthand way of expressing repeated multiplication of the same number. It consists of two parts: a base (the number being multiplied) and an exponent or power (which indicates how many times the base is multiplied by itself). For example, in 5³, 5 is the base and 3 is the exponent.
2. How is number power expressed as a formula or expression?
A number power is expressed as aⁿ, where 'a' is the base and 'n' is the exponent. This expression means you multiply 'a' by itself 'n' times. For instance, 4⁵ (read as '4 to the power of 5') is expressed as 4 × 4 × 4 × 4 × 4, which equals 1024.
3. What is the result when any number is raised to the power of 1 or 0?
There are two specific rules for exponents 1 and 0:
- Any number raised to the power of 1 is the number itself (e.g., 9¹ = 9).
- Any non-zero number raised to the power of 0 is always 1 (e.g., 9⁰ = 1).
4. Why does a negative exponent indicate the reciprocal of a number?
A negative exponent represents repeated division. Think of the pattern: 10² = 100, 10¹ = 10, 10⁰ = 1. Each time the exponent decreases by one, the result is divided by the base (10). Following this pattern, 10⁻¹ is 1 ÷ 10, which is 1/10. Therefore, a negative exponent like a⁻ⁿ is mathematically defined as the reciprocal of the positive exponent, or 1/aⁿ.
5. What are the basic laws of exponents used to simplify expressions?
To simplify calculations involving number powers, we use the laws of exponents. As per the CBSE/NCERT syllabus for 2025-26, the key laws are:
- Product Law: To multiply powers with the same base, add the exponents (aᵐ × aⁿ = aᵐ⁺ⁿ).
- Quotient Law: To divide powers with the same base, subtract the exponents (aᵐ ÷ aⁿ = aᵐ⁻ⁿ).
- Power of a Power Law: To raise a power to another power, multiply the exponents ((aᵐ)ⁿ = aᵐⁿ).
- Power of a Product Law: A power applied to a product is the same as the product of the powers ((ab)ⁿ = aⁿbⁿ).
6. How is the concept of number power applied in the real world?
The most common real-world application of number power is scientific notation. Scientists and engineers use powers of 10 to express very large or very small numbers concisely. For example, the distance from Earth to the Sun (approx. 150,000,000 km) can be written as 1.5 × 10⁸ km. Similarly, the width of a human hair can be written using negative powers.
7. What does a fractional exponent, like 9¹/², represent?
A fractional exponent represents a root of a number. An exponent of 1/2 means the square root, 1/3 means the cube root, and so on. Therefore, 9¹/² is the same as the square root of 9 (√9), which is 3. This concept connects powers directly to radicals and roots in algebra.
8. Is there a difference between (-2)⁴ and -2⁴?
Yes, there is a crucial difference. In (-2)⁴, the base is -2, so the expression means (-2) × (-2) × (-2) × (-2), which equals a positive 16. In -2⁴, the exponent applies only to the base 2, not the negative sign. So, it means -(2 × 2 × 2 × 2), which equals -16. The parentheses are very important for determining the base.
