Rational Exponents and Radicals

Write the following expression in radical form:

• a.)

• b.)

• c.)

• d.)

Answer explanation

The expression in radical form corresponds to choice c.), which correctly represents the original expression using a square root or other radical notation. The other options do not accurately reflect this transformation.

Write the following expression in exponential form:

• a.)

• b.)

• c.)

• d.)

Answer explanation

The expression is correctly written in exponential form as shown in option a.), which follows the standard notation for exponents. The other options do not represent the expression accurately.

.

• A

• B

• C

• D

Answer explanation

Choice A is correct because it directly addresses the question's requirements, while the other options do not provide the necessary information or context needed to answer effectively.

Rewrite in Exponential Form.

• A

• B

• C

• D

Answer explanation

To rewrite in exponential form, identify the base and the exponent. The correct choice D represents this transformation accurately, showing the relationship between the base and the exponent clearly.

• A

• B

• C

• D

Simplify.

• -25

• -5

• 1/25

• 1/5

Answer explanation

To simplify, divide -5 by -25. The negatives cancel out, resulting in 1/5. Thus, the correct answer is 1/5.

Answer explanation

To convert the radical expression \(\sqrt[3]{x^5}\) to exponent form, use the rule \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\). Here, \(m=5\) and \(n=3\), so it becomes \(x^{\frac{5}{3}}\). Thus, the correct answer is \(x^{\frac{5}{3}}\).

Answer explanation

To simplify the expression (x^{1/2} \cdot x^{3/2}), use the property of exponents that states a^m \cdot a^n = a^{m+n}. Here, 1/2 + 3/2 = 5/2, so the simplified expression is x^{5/2}.

Answer explanation

To rewrite \(\sqrt[4]{y^3}\) using rational exponents, we express it as \(y^{\frac{3}{4}}\). The denominator of the exponent (4) represents the root, and the numerator (3) represents the power.

Answer explanation

To simplify \((a^{\frac{2}{3}})^3\), use the power of a power property: multiply the exponents. Thus, \(\frac{2}{3} \times 3 = 2\). Therefore, the expression simplifies to \(a^2\), which is the correct answer.