Rewrite $\sqrt[3]{r^{11}}$ in simplest form.
Answer (1)
Rewrite the radical using a fractional exponent: 3 r 11 = r 3 11 .
Decompose the exponent: r 3 11 = r 3 + 3 2 .
Separate the exponents: r 3 + 3 2 = r 3 ⋅ r 3 2 .
Convert back to radical form: r 3 ⋅ r 3 2 = r 3 3 r 2 . The simplest form is r 3 3 r 2 .
Explanation
Understanding the problem We are asked to rewrite the expression 3 r 11 in its simplest form. This involves extracting any perfect cube factors from the radicand.
Converting to fractional exponent First, let's rewrite the radical using a fractional exponent: 3 r 11 = r 3 11
Decomposing the exponent Now, we want to decompose the exponent 3 11 into an integer and a proper fraction. We can write 3 11 as 3 + 3 2 since 11 = 3 × 3 + 2 . Therefore, we have: r 3 11 = r 3 + 3 2
Separating the exponents Using the properties of exponents, we can rewrite this as a product: r 3 + 3 2 = r 3 ⋅ r 3 2
Converting back to radical form Finally, we convert the fractional exponent back into a radical: r 3 ⋅ r 3 2 = r 3 ⋅ 3 r 2 Thus, the simplified form of the expression is r 3 3 r 2 .
Final Answer Therefore, the simplest form of 3 r 11 is r 3 3 r 2 .
Examples
Imagine you are calculating the volume of a peculiar shaped crystal. The crystal's volume can be expressed as 3 x 11 , where x is a measure of its side length. To better understand and calculate this volume, you simplify the expression to x 3 3 x 2 . This simplified form makes it easier to compute the volume for different side lengths, allowing you to quickly assess the crystal's size and properties.
