Exponents Study Guide (page 2)

Introduction to Exponents

I was x years old in the year x².

—Augustus De Morgan (1806–1871) British Mathematician

In this lesson, you'll learn how to work with terms that have positive and negative exponents, and we'll introduce fractional exponents.

Previously you learned that you can only add and subtract terms that have like bases and exponents. You also learned how to multiply terms. After multiplying the coefficients, if two terms have the same base, you added their exponents. In that same lesson, you learned that when dividing terms, if the terms had the same base, you subtract the exponent of the divisor from the exponent of the dividend. What else can we do with exponents? For starters, we can raise a base with an exponent to another exponent.

You already know that x⁴ means (x)(x)(x)(x). But what does (x⁴)² mean? The term x⁴ is raised to the second power. When a term is raised to the second power, or squared, we multiply it by itself: (x⁴)² = (x⁴)(x⁴). Now we have a multiplication problem. Keep the base and add the exponents: 4 + 4 = 8, so (x⁴)² = (x⁴)(x⁴) = x⁸.

What does (x³)⁶ mean? It means multiply x³ six times: (x³)(x³)(x³)(x³)(x³)(x³). Adding the exponents, we can see that (x³)⁶ = x¹⁸. Is there an easier way to find (x³)⁶, without having to write out a long multiplication sentence? Yes. When raising a base with an exponent to another exponent, multiply the exponents.

Look again at (x³)⁶: (3)(6) = 18, and (x³)⁶ = x¹⁸.

If the term has a coefficient, raise the coefficient to the exponent, too. To find (4k²)⁷, find (47)(k²)⁷: (47) = 16,384. (2)(7) = 14, so (k²)⁷ = k¹⁴. (4k²)⁷ = 16,384k¹⁴.

Be sure to look carefully at a term before multiplying exponents. In the expression (5x⁵)⁸, the entire term, coefficient and base, is raised to the eighth power. In the expression 6(a⁴)⁵, only the base and its exponent, a⁴, is raised to the fifth power. Why? Remember the order of operations. In (5x⁵)⁸, 5x⁵ is in parentheses, which means do multiplication in parentheses before the term is raised to the eighth power. Because we cannot simplify 5x⁵ any further, raise the entire term to the eighth power. In the expression 6(a⁴)⁵, the 6 is outside the parentheses. Exponents come before multiplication in the order of operations, so raise a⁴ to the fifth power before multiplying by 6.

If a term contains more than one variable raised to an exponent, each variable must be raised to the exponent. To find (a²b²)⁶, raise both a² and b² to the sixth power. (a²)⁶ = a¹² and (b²)⁶ = b¹², so (a²b²)⁶ = a¹²b¹².