Exponent Laws Made Easy: Step-by-Step Tutorial

Welcome to our lesson on exponent rules! Today we'll explore multiplication and division with exponents.When multiplying terms with the same base, we can add their exponents. Let's look at the general rule.Let's break down an example: two to the third power times two to the fourth power.We could write this out the long way, multiplying all the twos together.But using our rule, we can simply add the exponents: three plus four.This gives us two to the seventh power.Which equals one hundred and twenty-eight.Now let's look at division with exponents. When dividing terms with the same base, we subtract the exponents.Let's see how this works with x to the fifth power divided by x squared.Written out, we're dividing five x's by two x's.Using our rule, we subtract the exponents: five minus two.This simplifies to x cubed.Let's try a more challenging example with larger numbers.Consider five to the fourth power times five cubed.Using our multiplication rule, we add the exponents.This gives us five to the seventh power.Which equals seventy-eight thousand, one hundred and twenty-five.When we raise a power to another power, we multiply the exponents.Let's compare two ways to solve the same problem: two cubed, squared.Using our power rule makes the calculation more straightforward.This rule works with any base. Here's an example with variable x.Now let's explore the zero exponent rule. Any non-zero number raised to the power of zero equals one.We can understand why this is true using the division rule of exponents.When we divide x to the fifth power by itself, the exponents subtract, giving us x to the zero power, which equals one.This rule works for any non-zero number. Here are some examples.Let's try one more example to practice the power of power rule.Let's explore negative exponents, which give us a way to represent division using exponents.When we have a negative exponent, we can rewrite it as a fraction with 1 over the base raised to the positive exponent.Let's try another example. Three to the negative two becomes one over three squared, which equals one ninth.We can visualize negative exponents on a number line. Notice how positive and negative exponents are reciprocals of each other.Now let's explore fractional exponents, which give us a way to represent roots using exponent notation.A fractional exponent with just a one in the numerator represents a root. The denominator tells us which root to take.When we have a fraction with a numerator other than one, we first take the root, then raise to the power in the numerator.For example, sixteen to the three fourths means we take the fourth root of sixteen, which is two, then cube that result to get eight.Let's review the key points about negative and fractional exponents.Remember: Negative exponents create reciprocals. A fractional exponent with just a one on top means take a root. And when there's a different number on top, take the root first, then raise to that power.Thanks for learning about exponents with Spark.E!