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.
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.
Solved Examples
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
Practice Questions
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.