Exponents and powers play a crucial role in mathematics, allowing us to express large numbers efficiently and solve problems involving repeated multiplication. Exponential numbers refer to numbers expressed using an exponent or power. They involve a base number raised to an exponent.
The general form of an exponential number is : ab
where:
- a is the base,
- b is the exponent or power.
In this article, we will understand the concept of Exponents and Powers and solve some questions based on it.
What are Exponents?
Exponents are a way to show that a number is multiplied by itself a certain number of times. If you have an, a is the base and n is the exponent. It tells you to multiply a by itself n times. Exponents make it easier to write and work with large numbers or repeated multiplication.
For example, in the expression 23, 2 is the base, and 3 is the exponent. This means you multiply 2 by itself three times: 2 × 2 × 2 = 8.
Properties of Exponential
Exponential properties refer to the mathematical rules that govern the manipulation and simplification of expressions involving exponents. These properties are fundamental in algebra and are used to simplify and solve equations involving exponential expressions.
Given below is a summarized table of all the exponent properties that we have discussed above.
Property | Rule | Description | Example |
|---|---|---|---|
Product of Powers | am⋅an = am+n | Multiplication of same bases, add exponents. | 23⋅24 = 23+4 = 27= 128 |
Quotient of Powers | am/an=am−n | Dividsion of same bases, subtract exponents. | 56/52 = 56−2 =54 = 625 |
Power of a Power | (am)n=am⋅n | Raise an expression to another power, multiply the exponents. | (32)3 = 32⋅3 = 36 = 729 |
Power of a Product | (ab)n=an⋅bn | Raise a product to a power, raise each factor to the power separately. | (2⋅4)3 = 23⋅43 = 8⋅64 = 512 |
Power of a Quotient | (a/b)n=an /bn | Raise a quotient to a power, raise both numerator and denominator to the power. | (3/2)2 = 32/22 = 9/4 |
Zero Exponent | a0 = 1 | Any non-zero base raised to the power of zero is 1. | 70 = 1 |
Negative Exponent | a−n = 1/an | Negative exponent indicates reciprocal of the base. | 4-2 = 1/42 = 1/16 |
Fractional Exponent | am/n= | Fractional exponent represents a root. Numerator is power, denominator is root. | 81/3 = |
Practice Questions on Exponents and Powers - Solved
These Practice Questions on Exponents and Powers will help you better understand and apply the concept.
Example 1. Simplify the expression 34⋅32
According to product of powers property: am⋅ an= am + n
Therefore, 34⋅32= 34+2
= 36
= 3×3×3×3×3×3
= 729
Example 2: Simplify the expression 107/103
According to quotient of powers property: am/an= am−n
Therefore, 107/103 = 107-3
= 104
= 10,000
Exampe 3: Simplify the expression (25)2
According to power of a power property: (am)n= a m⋅n
Therefore, (25)2 = 25·2
= 210
= 1024
Example 4: Simplify the expression 7−3
According to negative exponent property: a−n= 1/an
Therefore, 7 -3 = 1/73
= 1/343
Example 5: Simplify the expression: (42 × 4-3).
Using the product rule am × an= am+n
Now, by applying the rule in given expression 42×4-3 we get,
= 42+(-3)
= 4-1
Now the outcome is in the form of negative exponent rule a-n = 1/an.
∴ 4-1 =1/41
Therefore, the simplified expression is 1/4
Example 6: Simplify the expression: (a3b2)3
This questions involves the power of a product rule (ab)n and the power of a power rule (am)n.
First we will apply the power of a product rule:
Distribute the outer exponent to both a3 and b2.
In the given expression (a3b2)3
(a3b2)3=(a3)3×(b2)3
Now, apply the power of a power rule to each term:
⇒ (a3)3=a3×3=a9
⇒ (b2)3=b2×3=b6
Combining the results we get,
(a3b2)3=a9× b6
Example 7: Simplify the expression:
23=2×2×2
=8
Rewrite the expression with the simplified numerator
(8/5)2
= 82/52
82 = 64
52 = 25
We can write it as = 64/25
Example 8: Solve for y: 52y+1 = 125
Given, 52y+1 = 125
We know, 125 = 53
Hence, 52y+1 = 53
Therefore, 2y+1 = 3
= 2y = 3-1 = 2
= 2y = 2
Hence y = 1
Example 9: Solve
Given
\left( \frac{a^4 b^2}{a^{-1} b^3} \right)^{-2} On solving powers, we get
(a4-(-1) b -2-3)-2
= (a5b-5)-2
= a-10 b10
= b10/a10
= (b/a)10
Example 10: Evaluate (81/3 × 161/4)
We have, (81/3 × 161/4)
By Fractional Exponent Property we have,
81/3 =
\sqrt[3]{8} = 2 and 161/4 =
\sqrt[4]{16} = 2 Hence the given equation becomes:
81/3 × 161/4 = 2 × 2 = 4
Read More:
- Adding and Subtracting Exponents
- Laws of Exponents
- How to Multiply and Divide Exponents
- Fractional Exponents
Practice Questions on Exponents and Powers - Unsolved
1. Simplify the expression: 53×52.
2. Simplify the expression:
3. Simplify the expression: (34)2.
4. Simplify the expression: (2×7)3.
5. Simplify the expression: (4/5)2.
6. Simplify the expression: 90.
7. Simplify the expression: 7−3.
8. Simplify the expression: (x2×y3)4.
9. Simplify the expression:
10. Simplify the expression: 103×23.
