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Exponents are subject to many laws, just like other mathematical properties. These are X^1 = X, X^0 = 1, X^-1 = 1/X, X^m * X^n = X^m+n, X^m/X^n = X^m-n, (X^m)^n = X^(m*n), (XY)^n = X^n * Y^n, (X/Y)^n = X^n/Y^n, and X^-n = 1/X^n.

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Q: What are different laws of exponents?

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The laws of exponents work the same with rational exponents, the difference being they use fractions not integers.

the base and the laws of exponent

They are not. Exponents, powers and indices are terms used for the same thing.

That depends how you choose to number the laws.

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The laws of exponents work the same with rational exponents, the difference being they use fractions not integers.

the base and the laws of exponent

There is only one law for exponents in division, and that is 1/ax = a-x

Exponents are used in many different contexts and for different, though related, reasons. Exponents are used in scientific notation to represent very large and very small numbers. The main purpose it to strip the number of unnecessary detail and to reduce the risk of errors. Exponents are used in algebra and calculus to deal with exponential or power functions. Many laws in physics, for example, involve powers (positive, negative or fractional) of basic measures. Calculations based on these laws are simper if exponents are used.

They are not. Exponents, powers and indices are terms used for the same thing.

a2 X a6 = a8

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That depends how you choose to number the laws.

You can have an infinite number of different exponents on a base number, you would then have an infinite amount of different numbers.

This is one of the laws of exponents, which states that xa * xb = x(a+b) The base is x, and the two powers (or exponents) are a and b.

You do not. The exponent is only subtracted in division.

Convert all expressions to the same base.