Exercise 142 Page 441

Practice makes perfect

Let's consider the given expression. 3^5/3^(10) We want to simplify the expression. We can do this by using the Quotient of Powers Law. This law states that to divide powers with the same base, we can subtract their exponents.

3^5/3^(10)

DivPow

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

3^(5-10)

SubTerm

Subtract term

3^(- 5)

To give the answer without negative exponents, we will use the Negative Exponent Property.

Negative Exponent Property
a^(- n)=1/a^n, for every nonzero number a

Let's rewrite our expression using this property.

3^(- 5)

NegExponentToFrac

a^(- m)=1/a^m

1/3^5

We want to simplify the following expression. 10x^4(10x)^(- 2) Let's start by rewriting (10x)^(- 2) using the Negative Exponent Property.

10x^4(10x)^(- 2)

NegExponentToFrac

a^(- m)=1/a^m

10x^4* 1/(10x)^2

Next, we will simplify the fraction using the Power of a Product Law. This law states that if we want to calculate the power of a product, we distribute the power to each factor and then multiply.

10x^4* 1/(10x)^2

PowProdII

(a b)^m=a^m b^m

10x^4* 1/10^2 x^2

From here, let's continue simplifying using the Quotient of Powers Law and the Negative Exponent Property.

10x^4* 1/10^2 x^2

MoveLeftFacToNumOne

a* 1/b= a/b

10x^4/10^2 x^2

WriteProdFrac

Write as a product of fractions

10/10^2* x^4/x^2

a=a^1

10^1/10^2* x^4/x^2

DivPow

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

10^(1-2)* x^(4-2)

SubTerms

Subtract terms

10^(- 1)* x^2

NegOneExponentToFrac

a^(- 1)=1/a

1/10 * x^2

MoveRightFacToNumOne

1/b* a = a/b

x^2/10

Let's consider the given expression. (1/4)^3* (4)^2 We want to simplify the expression. We can start by rewriting ( 14)^3 using the Negative Exponent Property.

(1/4)^3* (4)^2

FracToNegExponent

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

4^(- 3)* 4^2

Next, we will use the Product of Powers Law. Recall that the Product of Powers Law states that to multiply powers with the same base, we can add their exponents.

4^(- 3)* 4^2

MultPow

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

4^(- 3+2)

AddTerms

Add terms

4^(- 1)

Finally, we will use the Negative Exponent Property again to write our answer without negative exponents.

4^(- 1)

NegOneExponentToFrac

a^(- 1)=1/a

1/4