Question

Express the following radical form.1.5¾2.(3xy)⅔3.6a³/54.b7/45.(X+y)⅓​

Answer 1

Let's express each of the given radical forms step-by-step.

1. \( 1.5^{\frac{3}{4}} \)
This expression represents the three-fourths power of 1.5. In radical form, this would be the fourth root of 1.5 raised to the power of 3, which can be written as:
\[ \sqrt[4]{1.5^3} \]
or in other words, you would take 1.5, cube it, and then take the fourth root of the result.

2. \( (3xy)^{\frac{2}{3}} \)
This expression represents the two-thirds power of the product \(3xy\). In radical form, this would be the cube root of \( (3xy)^2 \), which can be written as:
\[ \sqrt[3]{(3xy)^2} \]
This means you square the product \(3xy\), and then take the cube root of the squared value.

3. \( 6a^{\frac{3}{5}} \)
This expression represents the three-fifths power of 6a (where "a" is raised to the third power). However, since the exponent applies only to "a" and not to the coefficient "6", the radical expression should be corrected as follows:
\( (6 \cdot a^3)^{\frac{1}{5}} \) which can be written as:
\[ \sqrt[5]{6a^3} \]
This means you take the fifth root of the product of 6 and \(a^3\).

4. \( b^{\frac{7}{4}} \)
This expression represents the seven-fourths power of b. In radical form, this can be expressed as the fourth root of \( b^7 \), which can be written as:
\[ \sqrt[4]{b^7} \]
This means you take the fourth root of \( b \) raised to the seventh power.

5. \( (x + y)^{\frac{1}{3}} \)
This expression represents the one-third power of the sum of x and y. In radical form, this can be expressed as the cube root of \( x + y \), which is written as:
\[ \sqrt[3]{x + y} \]
This means you take the cube root of the sum \( x + y \).

These are the expressions in radical form, where the rational exponents have been converted into roots.

Answer 2

The expressions involve converting radical terms into their exponential form, which include converting 1.5³⁴ to (1.5)^(3/4), (3xy)Ⅲ to (3xy)^(2/3), 6a³/5 to (6a^3)^(1/5), b7/4 to b^(7/4), and (X+y)⅓ to (X+y)^(1/3).

Step-by-step explanation:

The question asks for the expression of various radical terms in exponential form. Using the property of exponentials, we can convert radicals to their corresponding fractional exponents, which facilitates their simplification and manipulation. Below are the conversions of each given radical expression:

• For 1.5³⁴, we write it as (1.5)^(3/4).
• (3xy)Ⅲ becomes (3xy)^(2/3).
• For 6a³/5, the conversion is (6a^3)^(1/5).
• b7/4 is expressed as b^(7/4).
• (X+y)⅓ is written as (X+y)^(1/3), often referred to as the cube root of (X+y).

These conversions use the principle that the nth root of a number x can be written as x^(1/n) and that when a power is raised to another power, the exponents are multiplied. This forms the basis for expressing radicals as exponential terms.