Pre-Algebra: Exponent Properties Objective Simplify expressions using the properties of exponents. Exponents are used to simplify expressions. For example, a*a*a*a*a*a*a is the expanded expression of a 7. Exponents represent repeated multiplication of numbers or variables. For example, a 3 *a 2 is the same as a*a*a)**a*a). Add up the number of a s, it is now a 5. A better way to arrive at a solution is to use the product rule. The product rule of exponents states that if the bases are the same, you add the exponents. a m *a n = a m+n Example: 3 3 *3 4 *3: Are the bases the same? Yes, so add up the exponents, which are 3+4+1 = 8. 3 3 *3 4 *3 = 3 8 x 7 *y 5 *x 6 * y 3 : Use the product rule to simplify this expression by grouping all the same bases together. x 7 *x 6 *y 5 *y 3 = x 13 y 8 3xy 4 z 3 *5x 2 yz 8 : Group like bases together 3*5*x *x 2 * y 4 * y * z 3 * z 8 : Simplify the expression 15x 3 y 5 z 11 5*5 2 *5 7 : All the bases are the same. Add the exponents 5 1+2+7 = 5 10 6a 4 b 3 c 7 *2ab 2 d: Combine like terms and group like bases together 6*2*a 4 ab 3 b 2 c 7 d: Simplify 12a 5 b 5 c 7 d a 2 *b 3 *c 6 *a 3 *b 4 *c 5 : Group like bases together a 2 a 3 b 3 b 4 c 6 c 5 : Simplify the expression a 5 b 7 c 11
Divide using exponents. = Write the numerator and denominator in expanded form : Cancel out a y in the numerator by a y in the denominator. You are left with y*y or y 2. If the product rule states you add the exponents together when the bases are the same, what s going on when you divide the same variable with exponents? You subtract the exponents. Quotient Rule of Exponents states when dividing the same bases or variables with exponents, you subtract the exponents. = a m-n = 6 7-3 = 6 4 : Write each term separately and subtract the exponents with the same base or variable. For fractions, reduce if possible. x 3-1 y 5-4 z 8-5 : Subtract exponents to simplify x 2 yz 3 : Write the whole numbers as a fraction and subtract the exponents with the same base. a 7-4 b 5-3 c 2-1 = 1a 3 b 2 c = x 4-2 y 7-3 z 2-1 = x 2 y 4 z
We will now look at examples where the exponent is raised to a second exponent. For example, a 3 ) 2. This means the term are inside the parenthesis is being multiplied together 2 times a 3 )a 3 ). Because the bases are the same, we add the exponents, so the answer is a 6. A better method to find the answer is to use the Power of a Power Rule of exponents. a m ) n = a m*n Examples: 3 5 ) 2 : Each term inside the parenthesis is being multiplied together 2 times. The answer is 3 10. 4 4 ) 3 : Rewrite as 4 4*3 = 4 12 The next property we will look at is the Power of a Product Rule of Exponents. Use this when the bases are not the same. Please note that the power of a product rule is only used when the terms inside the parenthesis is being multiplied. It does not work if there is addition or subtraction. For example, a + b) 2 a 2 + b 2 ab) m = a m b m ab) 4 = a 4 b 4 2mn) 4 : Apply the exponent to each term inside the parenthesis. 2 4 m 4 n 4 = 16m 4 n 4 xy) 6 : Apply the exponent to each term inside the parenthesis x 6 y 6 3z 3 ) 3 : Apply the exponent to each term inside the parenthesis 3 3*1 z 3*3 : Use power to a power rule and simplify 27z 9 4a 4 b 3 c 2 ) 2 : Apply the exponent to each term inside the parenthesis 4 2 a 4*2 b 3*2 c 2*2 : Simplify using the power to a power rule 16a 8 b 6 c 4 m 4 n 2 *3mn 3 ) 2 : Per order of operations, simplify terms inside parenthesis first using product rule 3m 4 m 1 *n 2 n 3 ) 2 : Apply product rule 3m 5 *n 5 ) 2 : Apply the exponent to each term inside the parenthesis 3 2 m 5*2 n 5*2 : Simplify 9m 10 n 10
g 4 h 2 *4g 3 h) 3 : Per order of operations, simplify terms inside parenthesis first using product rule 4*g 4 g 3 *h 2 h) 3 : Simplify the terms 4g 7 * h 3 ) 3 : Use the Power of a Power rule 4 3 *g 7*3 *h 3*3 : Simplify 64g 21 h 9 x 3 *y 4 *3x 2 *y 3 ) 2 : Using order of operations, simplify terms insides parenthesis first using product rule 3x 3 x 2 *y 4 y 3 ) 2 : Apply product rule 3x 5 *y 7 ) 2 : Apply the exponent to each term inside the parenthesis 3 2 *x 5*2 *y 7*2 : Simplify 9x 10 y 14 We now use the property of exponents for fractions. This is called the Power of a Quotient Rule of Exponents. m = 3 : Multiply each exponent inside parenthesis by exponent outside the parenthesis. 2 : Multiply each exponent inside parenthesis by exponent outside the parenthesis. Special rule of exponents: Zero property of exponents : Using the quotient rule of exponents, we subtract the exponents a 3 3 = a 0 : How do we solve this example? Write the exponents as repeated multiplication = 1 : If you reduce all the a s, you have The result is a 0 = 1. This property is known as the zero property of exponents, which states that any number or expression raised to the zero power will always be 1.
Example: 3y 3 ) 0 = 1 4x 4 *y 3 z 9 ) 0 = 1 2a 0 b 4 ) 4 : Use zero property of exponents 2b 4 ) 4 : Apply exponent to each term inside parenthesis 2 4 b 4 = 16b 4 Now that we ve covered properties of exponents, we can mix up some examples: 4x 3 y 2 * 2x 4 y 3 ) 3 : First step is to simplify the terms inside the parenthesis using the product rule. 8x 7 y 5 ) 3 : Now use the power to a power rule to simplify the exponents. 8 3 x 7*3 y 5*3 : Evaluate 512x 21 y 15 7a 3 2a 4 ) 3 : Follow order or operations and use the applicable power rule. 7a 3 2 3 a 4*3 ): Simplify terms inside the parenthesis. 7a 3 8a 12 ): Use the product rule to find the solution. 7*8)a 3 *a 12 ): Simplify 56a 15 = Simplify the numerator and denominator using the product rule. : Simplify : Use quotient rule 8m 4-3 *n 4-4 : Simplify 8m : Simplify using the quotient rule x 4-2 y 3-3 : Simplify x 2
: Use the product rule for terms in the numerator : Multiply and simplify using product rule : Reduce the fraction and use quotient rule x 4-3 y 7-5 : Simplify 6xy 2 : Use the power of a power rule in the denominator : Simplify exponents in denominator : Use the quotient rule 3m 8-6 n 12-9 : Simplify 3m 2 n 3 ): Apply exponent to terms in innermost set of parenthesis ): Simplify exponents inside innermost parenthesis ): Simplify the numerator using the product rule. : Simplify whole numbers and exponents : Use quotient rule to simplify terms inside parenthesis * a 13-5 *b 8-7 ): Simplify 4a 8 b
4r 4 r 2 s 5 ) 2 : Use the power to a product rule. 4r 4 r 2*2 s 5*2 ): Simplify 4r 4 r 4 s 10 ): Simplify using the product rule. 4r 4+4 s 10 : Simplify 4r 8 s 10 : Use the power to the product rule to simplify the numerator : Simplify exponents : Use the quotient rule and simplify the fraction *p 6-3 *q 10-7 : Simplify *p 3 *q 3 or ) 2 : Use the product rule for terms inside the parenthesis ) 2 : Simplify numerator and apply exponent to each term in denominator 2 : Simplify terms in the denominator 2 : Use quotient rule to simplify terms in numerator and denominator m 3-2 *n 6-6 *p 4-2 ) 2 mp 2 ) 4 : Apply exponent to each term inside the parenthesis m 4 n 2*4 : Simplify m 4 p 8