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What Are the Laws of Exponents? Plus Examples

laws of exponents: Properties of exponents against a green background

The laws of exponents, sometimes known as the rules of exponents, are the rules and properties of exponents you use when simplifying or evaluating expressions containing exponents. The laws of exponents include definitions for what fractional exponents, negative exponents, and even different exponential expressions mean.

Remember, exponents are just a way of simplifying how many times a number is multiplied by itself. A number multiplied by itself n times can be denoted with an exponent n. This is called raising a number to its nth power. For example, 3 x 3 x 3 x 3 x 3 x 3 is the same as 3⁶, or three raised to the sixth power.

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What Is the Definition of a Linear Expression?

linear expression: equations on a grid paper

A linear expression is an algebraic statement where each term is either a constant or a variable raised to the first power. In other words, none of the exponents can be greater than 1.

For example, is a variable raised to the second power, but x is a variable raised to the first power.

5 is an example of a constant.

Note that the coefficients in front of the variables don't matter. Let's take a couple of these polynomials (a polynomial just means an expression with two or more terms) as examples.

2x - y + 3 is a linear expression.

x + y + z⁵ is a non-linear expression. It contains a term raised to the fifth power.

4 - 2 is a linear expression.

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What Is the Distributive Property of Multiplication?

distributive property of multiplication: girl taking notes while looking at her laptop

The distributive property of multiplication is a property of real numbers that shows how we can break apart multiplication problems into separate terms. The property states that an algebraic expression a(b + c) becomes ab + ac. In other words, the multiplication of a distributes to both variables inside the parentheses, b and c.

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Why Is the Square Root of Pi an Irrational Number?

square root of pi: Symbol and value of pi against a galaxy background

Have you ever been asked what the square root of pi is? Disclaimer: The quick answer is that there is no precise answer! Because all square roots of irrational numbers are irrational numbers, the square root of pi is also an irrational number. However, that doesn't mean we can't approximate the answer. Just like we approximate the value of pi to be 3.14, we can approximate the square root of pi to be 1.77.

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Greatest Common Factor Definition: What You Need to Know

greatest common factor definition: Numbers written using a brush

The greatest common factor (GCF) is a term used to describe the biggest number that can divide evenly into two or more numbers. Sometimes, this is also referred to as the greatest common divisor (GCD) or highest common factor (HCF). A factor is a smaller number that can divide evenly into that number. Let’s look at the greatest common factor definition.

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