Exercise 114 Page &nbsp; 433

Commutative Property of Multiplication

4* 3* x^3* x* y* y^2

AssociativePropMult

Associative Property of Multiplication

(4* 3)* (x^3* x)* (y* y^2)

12* (x^3* x)* (y* y^2)

a=a^1

12* (x^3* x^1)* (y^1* y^2)

a^m*a^n=a^(m+n)

12* x^(3+1)* y^(1+2)

Add terms

12 x^4 y^3

Let's consider the given expression. 6a^5b^2* 3ab^2 To simplify this expression, we can use the Product of Powers Law. Like in Part A, we will also need to use the Commutative and Associative Properties of Multiplication. Let's do it!

6a^5b^2* 3ab^2

Commutative Property of Multiplication

6* 3* a^5* a* b^2* b^2

AssociativePropMult

Associative Property of Multiplication

(6* 3)* (a^5* a)* (b^2* b^2)

18* (a^5* a)* (b^2* b^2)

a=a^1

18* (a^5* a^1)* (b^2* b^2)

a^m*a^n=a^(m+n)

18* a^(5+1)* b^(2+2)

Add terms

18 a^6 b^4

We are asked to simplify the algebraic expression. m^2n* 9mn Again, we can do this using the Product of Powers Law and the Commutative and Associative Properties of Multiplication. Let's do it!

m^2n* 9mn

Commutative Property of Multiplication

9* m^2* m* n* n

AssociativePropMult

Associative Property of Multiplication

9* (m^2* m)* (n* n)

a=a^1

9* (m^2* m^1)* (n^1* n^1)

a^m*a^n=a^(m+n)

9* m^(2+1)* n^(1+1)

Add terms

9 m^3 n^2

Let's consider the given expression. 3^5 * 8* 5^3/3^2* 2^3 * 5^3* 3^3 We want to simplify the expression. We can do this by using the Product of Powers Law and the Quotient of Powers Law. Recall that the Quotient of Powers Law states that to divide powers with the same base, we need to subtract their exponents. In this case, we will also use the Commutative and Associative Properties of Multiplication.

3^5 * 8* 5^3/3^2* 2^3 * 5^3* 3^3

Commutative Property of Multiplication

3^5 * 8* 5^3/3^2* 3^3 * 2^3 * 5^3

a^m*a^n=a^(m+n)

3^5 * 8* 5^3/3^(2+3)* 2^3 * 5^3

Add terms

3^5 * 8* 5^3/3^5* 2^3 * 5^3

WriteProdFrac

Write as a product of fractions

3^5/3^5 * 8/2^3 * 5^3/5^3

CalcPow

Calculate power

3^5/3^5 * 8/8 * 5^3/5^3

QuotOne

a/a=1

3^5/3^5 * 1 * 5^3/5^3

DivPow

a^m/a^n= a^(m-n)

3^(5-5)* 1 * 5^(3-3)

SubTerms

Subtract terms

3^0 * 1 * 5^0

ExponentZero

a^0=1

1* 1 * 1

IdPropMult

Identity Property of Multiplication

We want to simplify the algebraic expression. m^4* n/n^3 We can start by factoring the expressions in both the numerator and denominator. Then we will cross out the common factors. Let's do it!

m^4* n/n^3

SplitIntoFactors

Split into factors

m * m * m * m * n/n* n* n

CancelCommonFac

Cancel out common factors

m * m * m * m * n/n* n* n

SimpQuot

Simplify quotient

m * m * m * m/n* n

m^4/n^2

Let's analyze the given expression. 9a^4 b^2/15b To simplify the expression, we can start by factoring the expressions in both the numerator and denominator. Then we will cross out the common factors.

3* 3* a * a * a * a* b* b/3* 5 * b

CancelCommonFac

Cancel out common factors

3* 3* a * a * a * a* b* b/3 * 5 * b

SimpQuot

Simplify quotient

3* a * a * a * a* b/5