Exercise 67 Page 141

Practice makes perfect

When rewriting a radical into exponential form, the power of the radicand is the numerator of the rational exponent, and the index of the radical is the denominator of the rational exponent.

sqrt(a)=a^()1 n and sqrt(a^m)=a^() m n With this in mind, we can rewrite the given expression. &sqrt(x) ⇔ x^()1 5

Notice that the denominator of the given fraction has a positive exponent. When this is the case, the denominator can be moved to the numerator and the exponent will become negative.

1/a^(mn)=a^(- mn) ⇒ 1/x^3=x^(-3)

To simplify the given expression, remember that the numerator of a rational exponent is the power of the expression, and the denominator is the index.

a^()1 n=sqrt(a) and a^() m n=sqrt(a^m) Let's rewrite the given expression. x^(2 3)=sqrt(x^2)

When rewriting a radical into exponential form, the power of the radicand is the numerator of the rational exponent, and the index of the radical is the denominator of the rational exponent.

sqrt(a)=a^()1 n and sqrt(a^m)=a^() m nWith this in mind, we can rewrite the given expression. &1/sqrt(x) ⇔ 1/x^()1 2 Notice that the denominator of the given fraction has a positive exponent. When this is the case, the denominator can be moved to the numerator and the exponent will become negative. 1/a^(mn)=a^(- mn) ⇒ 1/x^(12)=x^(- 12)

To simplify the given expression we will use the Properties of Rational Exponents. Remember that when you have a negative exponent of an integer you can move it to the denominator and change the exponent into a positive number. Let's do it!

x^(-1)y^(-8)

NegExponentToFrac

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

1/x^1* 1/y^8

MultFrac

Multiply fractions

1/xy^8

To simplify the given expression we will use the Properties of Rational Exponents. For this exercise, we will begin by multiplying the exponents.

(m^2)^(- 32)

PowPow

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

m^(-2* 32)

DenomMultFracToNumber

2 * a/2= a

m^(-3)

Notice that the exponent is negative, so the number can be moved to the denominator and the exponent will become positive. a^(- m)=1/a^m ⇒ m^(-3)=1/m^3

To simplify the given expression, we will use the Properties of Rational Exponents. For this exercise, we will begin by distributing the exponent to both factors. Then we will try to simplify the exponents. Let's do it!

(x^3y^6)^()12

PowProd

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

(x^3)^()12(y^6)^()12

PowPow

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

x^(32)y^(62)

CalcQuot

Calculate quotient

x^(32)y^3

We can rewrite the denominator of the rational exponent as the index of the radical. x^() 3 2y^3=sqrt(x^3)y^3

To simplify the given expression we will use the Properties of Rational Exponents. For this exercise, because the rational exponent is negative we will need to use the fact that a negative exponent becomes positive when written on the other side of a fraction bar.

(9x^3y^6)^(- 2)=1/(9x^3y^6)^2 Now, let's distribute the exponent to the terms inside the parentheses.

1/(9x^3y^6)^2

PowProd

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

1/9^2(x^3)^2(y^6)^2

PowPow

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

1/81x^6y^(12)