Multiplying exponents is an essential part of algebra, and many students face many problems while solving them; they don’t know how to multiply exponents. Algebra is an essential part of math and has many uses in higher studies.
Multiplying exponents is a little tricky to understand and solve equations of algebra. Still, everything works on some rules in math, we have to follow all the rules and practice them with examples, but we have to understand what it means. So, let’s first understand it.
When multiplying, the exponent represents the repetition multiplication of any number with itself, which is also expressed as power in math. It is represented as 5^6, which means 5x5x5x5x5x5, and we have to multiply it and get the result.
The first number, i.e. 5, is called the base, which will be multiplied by itself, and the second number, i.e. 6, is called the exponent of it, which means six times you have to multiply the number 5 by itself.
Now, come and see the rules of exponents
Before knowing how to multiply exponents, it’s crucial to understand the rules of exponents. These rules help in avoiding wrong calculations.
- Rule of the product of power:-
This Rule means you have to add exponent together while multiplying the same base number.
For example :- 5^3 and 5^2 = 5^(3+2) = 5^5
- Rule of Quotient of power:-
This Rule means you have to subtract the powers together while dividing the same base number.
For example :- 5^3 / 5^2 = 5^(3-2) = 5^1
- Rule of power to power:-
This Rule means you must multiply the power when raising to another exponent.
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For example :- (5^2)^2 = (25)^2
- Rule of quotient:-
This Rule means you must distribute all the power values in the quotient.
For example :- (4/5)^2 = (4×4 / 5×5)
- Rule of power zero:-
This Rule means if any numbers have zero power, it equals 1.
For example :- 5^0 = 1
- Rule of negative exponent:-
This Rule means if you want to change the negative power of any number, then do its reciprocal, i.e. if the number is in the numerator, send it to the denominator; if it is in the denominator, send it to the numerator.
For example :- 5^(-2) = (1/5 X 5)
These are some rules you must follow, and they will help you solve the exponent problem.
Now let’s see how to multiply exponents
1. Multiplying exponents with the same base:-
When the number has the same base and multiplies a different exponent, we have to add the power of the number to find the result where the base will be the same.
For example :- 5^3 and 5^2 = 5^(3+2) = 5^5
This is the shortcut to solve the problem when the base’s power keeps changing, but the base remains the same, and this Rule will be followed everywhere.
Another example :- (2x ^3 ) (3x^2) = (2x X 3x)^6
Here we will see first multiple the coefficient of x, which means any number multiplied by x and then add their power.
2. Multiplying exponents of different bases:-
When the bases are different, we have to multiply them, and the power will be the same.
For example :- 5^2 and 3^2 = (5X2)^2 = 10^2
This way of multiplication follows the Rule of distribution of all the power values in the quotient.
Another example :- (2x ^2 ) (3x^2) = (2x X 3x )^2
Here we will first multiply the coefficient of x, which means any number multiplied by x and then keep their power the same.
This Rule of multiplication can only be followed when the variable is the same, i.e. if only x or y, they can’t be mixed.
3. Multiplying exponents with different bases and power:-
When you have multiple bases and power, no shortcut can be followed because there is nothing common between them, so you have to do it manually by calculating them separately.
For example- 3^2 and 4^3
4. Multiplication of negative exponents:-
It may sound complicated, but it is the same as the others; you have to take care of the place or power of the base before multiplying it by any number.
For example :- 5^(-2) X 2^3= (2 X 2 X 2 /5 X 5)
5. Multiplication of base power zero:-
If any of the numbers in the base have power zero, and with the multiplication of the exponent with the different base, then that number will be equal to 1.
For example :- 5^(0) X 2^3= (2 X 2 X 2 X 1)
Here first, we have given the standard value of the number having power 0, i.e. one, and then multiplied it with the rest of the numbers.
6. Multiplying exponents with square root:-
This exponent’s multiplication is tricky, but the rules are the same. If any number has a square root, you can change radicals to rational exponents like -/a, which can be expressed as a^(1/2). Now, this can be multiplied by any power it has by rewriting it the same.
For example :- (-/2)^3 = 2^(1/2) X 3 = 2^(3/2)
Here first, we have to change the radical to rational exponent, i.e. 2^(1/2), and then multiply the power of base two by three, which will return as 2^(3/2)
7. Multiplication of exponents with fractions:-
When the base of the expression is in the fraction which has been raised to an exponent, then we have to follow the same Rule we have been following for a long time.
Some of the cases of base as a fraction:-
- When the exponent with different base fraction values is the same, we have to follow the Rule of the product of power and add them.
(a/b)^n X (a/b)^m = (a/b)^(n+m)
- When the exponent with the same and base fraction is different then multiple the base fractional number keeps the power the same.
(a/b)^n X (c/d)^n = (a/b X c/d)^n
- When the power and fractional base are both different, you have multiple numbers and the power.
(a/b)^n X (c/d)^m = (a^n X c^m) / ( b^n X d^m)
8. Multiplication of the exponent with fractional power:-
When the base has the fractional power, then we have to follow different Rule, which are
- When the exponent of the difference and base value is the same, we must follow the Rule of the product of power and add them.
a^(n/m) X a^(k/j) =a^( n/m) + a^(k/j)
- When the exponent is the same, and the base is different from multiple, the fractional base number keeps the power the same.
a^(n/m) X b^(n/m) =(a X b)^( n/m)
- When the power and fractional base are both different, you have multiple numbers and the power.
a^(n/m) X b^(k/j) =a ^( n/m) X b^(k/j)
9. Power raised to zero exponent multiplication:-
We know very well that 0 raised to power any number is always a zero, so the same Rule we have to follow here also.
For example- 0^2 and 0^3 it is always equal to 0
10. Multiplying exponents with different variables:-
You can use only multiple bases and add the power if they have the same variable because two different variables can’t be mixed.
For example :- (3x)^2 X (2x)^2 X (5y)^2 = (6x)^2 X (5y)^2
You may also like to read- Effective math teaching strategies
Activities for practicing multiplication of exponents:
1. Exponent maze property:-
This maze of exponents involves three in particular in which you will have to include a set of questions, which include a question on the topic covering all the properties of the exponent.
You must create three mazes of 15 questions, each with an answer key and one maze including a negative number.
2. Knockout board game:-
This knockout game will involve characters drawn on the whiteboard where an exponent question will hide each character. You have to solve the question of your chosen character; after solving all questions, the solution will be discussed.
3. Puzzle game:-
Many students love solving puzzle games, so it will be a great activity for them to learn and play with puzzles.
This puzzle will involve 12 questions of exponents in which six exponent’s properties will be used. You have to think and solve the expression by yourself and choose the answer from the square board of the puzzle.
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