An exponent indicates how many times the base number is multiplied by itself. It is written as small number to upper right of a Base number. The number that is being multiplied is called base. An exponent (or also called power) indicates how many times the number (the base) is multiplied by itself. For example in an , a is base and n is exponent.
For Example: In 53, 5 is base and 3 is the exponent. This means 5 × 5 × 5 = 125.
Multiplication of Exponents
The Multiplication of exponents refers to process of multiplying the numbers that have exponents. There are specific rules to follow depending on whether bases are same or different. General rule is that when multiplying two exponents with same base then add exponents together. When bases varies then special rules apply. Identify the bases and exponents: Determine base and the exponent of each term.
Multiplying Exponents with Same Base
When multiplying the terms with same base keep the base same and add the exponents together. This is key property in exponents known as The Product Rule. The general formula is:
am × an = a m + n
Example: 23 × 24
23 × 24 = 23 + 4 = 27 = 128
Multiplying Exponents with Different Base
When multiplying exponents with different bases but the same exponent, you can combine them using the exponent on the product of the bases.
am × bm = (a × b)m
Example
32 × 42 = (3 × 4)2 = 122 = 144
Multiplying Exponents with Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. When multiplying negative exponents with the same base, you can apply the rule of adding exponents.
a-n = 1/an
Example: 2-3 × 2-2 .
2-3 × 2-2 = 1/23 × 1/22 = 1/23+2 = 1/25 = 1/32
Power of a Power
If you are raising a power to another power, you multiply the exponents together.
(am)n = am×n
Example: (23)2 = 23×2 = 26 = 64
Multiplying Fractional Exponents
Fractional exponents represent roots as well as powers. When multiplying bases with fractional exponents, you still follow the basic exponent rules.
am/n × ap/q = am/n + p/q
Note: n√a = a1/n.
Example: √2 × √8.
√2 × √8 = 21/2 × 81/2 = (2 × 8)1/2 = 161/2 = 4
Summary
We can summarize all the rules in the following table:
| Concept | Formula |
|---|---|
| Multiplying Exponents with Same Base | am × an = a m + n |
| Multiplying Exponents with Different Bases | am × bm = (a × b)m |
| Multiplying Exponents with Negative Exponents | a-n = 1/an1 |
| Power of a Power | (am)n = am×n |
| Multiplying Fractional Exponents | am/n × ap/q = am/n + p/q |
Solved Questions on Multiplying Exponents
Q 1. Multiply 32 × 33
Solution:
Since bases are same so we can add the exponents.
Using same base rule: 32 × 33 = 32+3 = 35 =243
Q 2. Multiply 2-3 × 2-2
Solution:
Add exponents, if a negative exponent means the reciprocal so:
Adding exponents: 2-3 × 2-2 = 2-3-2 = 2-5 = 1/25 = 1/32
Q3. Simplify (x3)2
Solution:
Use power of power rule: (x3)2 = x3×2 = x6
Q4. Simplify 51/2 × 53/2
Solution:
With same base exponents can be added together,
Thus, 51/2 × 53/2 = 5(1/2) + (3/2) = 54/2 = 52 = 25
Q5. Multiply 32/43 × 22/52
Solution:
Multiply numerators and the denominators separately:
Combine and simplify: 32 × 22 / 43 × 52 = 9×4/64×25 = 36/1600 = 9/400
Q6. Multiply 72 × 7-4
Solution:
If exponent has negative number then it is reciprocal so:
Thus, 72 × 7-4 = 72-4 = 7-2 = 1/72 = 1/49
Q7. Multiply (23)4
Solution:
Using power of power rule, we get
(23)4 = 23×4 = 212 = 4096
Q8. Multiply x2 × y2
Solution:
Since bases are different then terms stay separate can not be combined.
Thus, x2 × y2 = (x × y)2
Worksheet on Multiplying Exponents
You can download free Worksheet on Multiplying Exponents from below:
Conclusion
In conclusion, multiplying exponents is a straightforward process once you understand the rules. When the bases are the same, you simply add the exponents. If the bases are different but have the same exponent, you can multiply the bases together and keep the exponent.
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