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- Introduction to Exponents

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$$x^n = x * x * x * x * x * x$$
note: There are n x's in the product.

x ≠ 0, x = base, n = exponent

Any number raised to the zero power (except 0) equals 1. $$x^0 = 1$$ Any number raised to the power of one equals itself. $$x^1 = x$$ To multiply terms with the same base, add the exponents. $$x^a * x^b = x^{(a+b)}$$ To divide terms with the same base, subtract the exponents. $$\frac{x^a}{x^b} = x^{(a-b)}$$ When a product has an exponent, each factor is raised to that power. $${(x^a)^b} = x^{a * b}$$ A number with a negative exponent equals its reciprocal with a positive exponent. $$x^{(-a)} = \frac{1}{x^a}$$ When a product of two numbers has an exponent, each factor is raised to that power. $${(x * y)^a} = x^a * y^a$$ An absolute value of the squared number equals squared value of the number. $$|x^2| = |x|^2 = x^2$$

A general example to help you recognize patterns and spot the information you're looking for

$$4^3 = 4 * 4 * 4$$ $$x^2 = x * x$$ $$5^0 = 1$$ $$7^1 = 7$$ $$3^2 * 3^4 = 3^{(2+4)} = 3^6 = 729$$ $$\frac{2^5}{2^3} = 2^{(5-3)} = 2^2 = 4$$ $$(3^2)^5 = 3^{(2*5)} = 3^{10} = 59049$$ $$10^{(-2)} = \frac{1}{10^2} = 0.01$$ $${(2*5)}^3 = 2^3 * 5^3 = 8 * 125 = 1000$$ $$|3^{(2)}| = |3|^2 = 3^2 = 9$$ $$|{(-3)}^{(2)}| = |{(-3)}|^2 = 3^2 = 9$$ Please remember: $${(-5)}^2 ≠ {-5}^2$$ $$(-5)^2 = {(-1)}^2 * 5^2 = 1 * 25 = 25$$ $${-5}^2 = {(-1)} * {(5)}^2 = {-1} * 25 = {-25}$$Watch a Khan Academy Video »

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- Last Updated: Apr 2, 2024 5:25 PM
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