Multiplying Exponents Calculator
Calculate multiplying exponents instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
a^m x a^n = a^(m+n) and a^n x b^n = (ab)^n
The Product of Powers rule states that when multiplying powers with the same base, you add the exponents. The Power of a Product rule states that when multiplying powers with the same exponent but different bases, you multiply the bases and keep the exponent. These are fundamental laws of exponents used throughout algebra.
Worked Examples
Example 1: Same Base Multiplication
Problem:Calculate 5^3 times 5^4 using the product of powers rule.
Solution:Same base (5), so add exponents: 5^(3+4) = 5^7\n5^7 = 5 x 5 x 5 x 5 x 5 x 5 x 5 = 78,125\nVerification: 5^3 = 125, 5^4 = 625\n125 x 625 = 78,125
Result:5^3 x 5^4 = 5^7 = 78,125
Example 2: Different Bases, Same Exponent
Problem:Calculate 4^3 times 7^3 using the power of a product rule.
Solution:Same exponent (3), so combine bases: (4 x 7)^3 = 28^3\n28^3 = 28 x 28 x 28 = 21,952\nVerification: 4^3 = 64, 7^3 = 343\n64 x 343 = 21,952
Result:4^3 x 7^3 = 28^3 = 21,952
Frequently Asked Questions
What is the rule for multiplying exponents with the same base?
When multiplying exponential expressions that share the same base, you keep the base unchanged and add the exponents together. This is known as the Product of Powers rule: a^m times a^n equals a^(m+n). For example, 2^3 times 2^4 equals 2^(3+4) which is 2^7 or 128. This rule works because exponents represent repeated multiplication. The expression 2^3 means 2 times 2 times 2, and 2^4 means 2 times 2 times 2 times 2. Multiplying them together gives seven factors of 2, which is 2^7. This rule applies to all real number bases and all integer and fractional exponents.
How do you multiply exponents with different bases?
When multiplying exponential expressions with different bases, you generally cannot simplify using the product of powers rule. Instead, you must evaluate each expression separately and then multiply the results. For example, 2^3 times 5^2 equals 8 times 25, which is 200. However, if the exponents are the same, you can use the Power of a Product rule: a^n times b^n equals (a times b)^n. So 2^3 times 5^3 equals (2 times 5)^3 which is 10^3 or 1000. This shortcut works because you are multiplying the same number of factors from each base.
Can exponents be negative when multiplying, and what does that mean?
Yes, negative exponents are fully valid and follow the same multiplication rules. A negative exponent means the reciprocal: a^(-n) equals 1 divided by a^n. When multiplying, you still add exponents: 3^2 times 3^(-5) equals 3^(2 + (-5)) which is 3^(-3) or 1/27. This concept is essential in scientific notation, where very small numbers are expressed using negative powers of 10. For instance, 0.001 is 10^(-3). Multiplying 10^4 by 10^(-7) gives 10^(-3). Negative exponents also appear in physics formulas, decay functions, and unit conversions.
What happens when you multiply exponents with fractional or decimal exponents?
Fractional exponents follow the same rules as integer exponents. The product of powers rule still applies: a^(1/2) times a^(1/3) equals a^(1/2 + 1/3) which is a^(5/6). Fractional exponents represent roots: a^(1/2) is the square root, a^(1/3) is the cube root, and a^(m/n) is the nth root of a raised to the mth power. For example, 8^(1/3) times 8^(2/3) equals 8^(1/3 + 2/3) which is 8^1, or 8. Decimal exponents like 2^(3.5) are equivalent to fractional forms such as 2^(7/2), and the same addition rules apply when multiplying.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy