Rules for Exponents - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST

Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

Chapter 7. Rules for Exponents

In Chapter 4, you learned about the various types of exponents that you might encounter in algebra. In this chapter, you learn about the rules for exponents—which you will find useful when you simplify algebraic expressions. The following rules hold for all real numbers x and y and all rational numbers m, n, and p, provided that all indicated powers are real and no denominator is 0.

Product Rule

Product Rule for Exponential Expressions with the Same Base

x^m xⁿ = x^m+n

This rule tells you that when you multiply exponential expressions that have the same base, you add the exponents and keep the same base.

If the bases are not the same, don’t use the product rule for exponential expressions with the same base.

Problem Simplify.

a. x² x³

b. x² y⁵

c. x² x⁷ y³ y⁵

Solution

a. x² x³

Step 1. Check for exponential expressions that have the same base.

x² x³

x² and x³ have the same base, namely, x.

Step 2. Simplify x² x³. Keep the base x and add the exponents 2 and 3.

x² x³ = x²⁺³ = x⁵

x²x³ ≠ x^2.3 = x⁶ When multiplying, add the exponents of the same base, don’t multiply them.

b. x²y⁵

Step 1. Check for exponential expressions that have the same base.

x²y⁵

x² and y⁵ do not have the same base, so the product cannot be simplified.

x²y⁵ ≠ (xy)⁷. This is a common error that you should avoid.

c. x²x⁷y³y⁵

Step 1. Check for exponential expressions that have the same base.

x²x⁷y³y⁵

x² and x⁷ have the same base, namely, x, and y³ and y⁵ have the same base, namely, y.

Step 2. Simplify x²x⁷ and y³y⁵ For each, keep the base and add the exponents.

x²x⁷y³y⁵ = x²⁺⁷y³⁺⁵ = x⁹y⁸

Quotient Rule

Quotient Rule for Exponential Expressions with the Same Base

This rule tells you that when you divide exponential expressions that have the same base, you subtract the denominator exponent from the numerator exponent and keep the same base.

If the bases are not the same, don’t use the quotient rule for exponential expressions with the same base.

Problem Simplify.

Solution

Step 1. Check for exponential expressions that have the same base.

x⁵ and x³ have the same base, namely, x.

Step 2. Simplify . Keep the base x and subtract the exponents 5 and 3.

. When dividing, subtract the exponents of the same base, don’t divide them.

Step 1. Check for exponential expressions that have the same base.

y⁵ and x² do not have the same base, so the quotient cannot be simplified.

. This is a common error that you should avoid.

Step 1. Check for exponential expressions that have the same base.

x⁷ and x² have the same base, namely, x, and y⁵ and y³ have the same base, namely, y.

Step 2. Simplify and For each, keep the base and subtract the exponents.

Step 1. Check for exponential expressions that have the same base.

x³ and x¹⁰ have the same base, namely, x.

Step 2. Simplify Keep the base x and subtract the exponents 3 and 10.

Step 3. Express x^–7 as an equivalent exponential expression with a positive exponent.

When you simplify expressions, make sure your final answer does not contain negative exponents.

Rules for Powers

Rule for a Power to a Power

(x^m)^p = x^mp

This rule tells you that when you raise an exponential expression to a power, keep the base and multiply exponents.

Problem Simplify.

a. (x²)³

b. (y³)⁵

Solution

a. (x²)³

Step 1. Keep the same base x and multiply the exponents 2 and 3.

(x²)³ = x^2.3 = x⁶

(x²)³ ≠ x⁵. For a power to a power, multiply exponents, don’t add.

b. (y³)⁵

Step 1. Keep the same base y and multiply the exponents 3 and 5.

(y³)⁵ = y^3.5 = 15¹⁵

Rule for the Power of a Product

(xy)^p = y^py^p

This rule tells you that a product raised to a power is the product of each factor raised to the power.

Confusing the rule (xy)^p = x^py^p with the rule x^mxⁿ = x^m+n is a common error. Notice that the rule (xy)^p = x^py^p has the same exponent and different bases, while the rule x^mxⁿ = x^m+n has the same base and different exponents.

Problem Simplify.

a. (xy)⁶

b. (4x)³

c. (x³y²z)⁴

Solution

a. (xy)⁶

Step 1. Raise each factor to the power of 6.

(xy)⁶ = x⁶x⁶

b. (4x)³

Step 1. Raise each factor to the power of 3.

(4x)³ = 4³x³ = 64x³

c. (x³y²z⁴)

Step 1. Raise each factor to the power of 4.

(x³y²z⁴ = (x)³)⁴(y)²)⁴(z)⁴ = x¹²y⁸z⁴

Rule for the Power of a Quotient

This rule tells you that a quotient raised to a power is the quotient of each factor raised to the power.

Problem Simplify.

Solution

Step 1. Raise each factor to the power of 4.

Step 1. Raise each factor to the power of 3.

Rules for Exponents Summary

You must be very careful when simplifying using rules for exponents. For your convenience, here is a summary of the rules.

Rules for Exponents

1. Product Rule for Exponential Expressions with the Same Base

x^mxⁿ = x^m+n

2. Quotient Rule for Exponential Expressions with the Same Base

3. Rule for a Power to a Power

(x^m)^p = x^mp

4. Rule for the Power of a Product

(xy)^p = x^py^p

5. Rule for the Power of a Quotient

Notice there is no rule for the power of a sum [e.g., (x + y)²] or for the power of a difference [e.g., (x – y)²]. Therefore, an algebraic sum or difference raised to a power cannot be simplified using only rules for exponents.

Problem Simplify using only rules for exponents.

a. (xy)²

b. (x + y)²

c. (x – y)²

d. (x + y)²(x + y)³

Solution

a. (xy)²

Step 1. This is a power of a product, so square each factor.

xy² = x²y²

b. (x + y)²

Step 1. This is a power of a sum. It cannot be simplified using only rules for exponents.

(x + y)² is the answer.

(x + y)² ≠x²+y²!(x + y)² = (x + y)(x + y) = x² + 2xy + y² (which you will learn in Chapter 9). This is the most common error that beginning algebra students make.

c. (x – y)²

(x – y)² ≠ x²-y²!

Step 1. This is a power of a difference. It cannot be simplified using only rules for exponents.

(x – y)² is the answer.

d. (x + y)²(x + y)³

Step 1. This is a product of expressions with the same base, namely, (x + y). Keep the base and add the exponents.

(x + y)²(x + y)³ = (x + y)²⁺³ = (x + y)⁵

When a quantity enclosed in a grouping symbol acts as a base, you can use the rules for exponents to simplify as long as you continue to treat the quantity as a base.

Step 2. (x + y)⁵ is a power of a sum. It cannot be simplified using only rules for exponents.

(x + y)⁵ is the answer.