Question Find the simplest form of $\sqrt[3]{4 x^{2}} \cdot \sqrt[3]{8 x^{7}}$ .

Studdy Solution

STEP 1

Assumptions
1. We are multiplying two cube roots together.
2. The properties of exponents and roots will be used to simplify the expression.
3. The goal is to express the product in its simplest form.

STEP 2

Recall the property of exponents that states when multiplying two powers with the same base, you add the exponents.
$a^m \cdot a^n = a^{m+n}$

STEP 3

Recall the property of radicals that states when multiplying two radicals with the same index, you can combine them under a single radical.
$\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$

STEP 4

Apply the property of radicals from STEP_3 to combine the two cube roots into one.
$\sqrt[3]{4 x^{2}} \cdot \sqrt[3]{8 x^{7}} = \sqrt[3]{4 x^{2} \cdot 8 x^{7}}$

STEP 5

Multiply the numerical coefficients and the variables with exponents separately.
$\sqrt[3]{4 \cdot 8} \cdot \sqrt[3]{x^{2} \cdot x^{7}}$

STEP 6

Calculate the product of the numerical coefficients.
$4 \cdot 8 = 32$

STEP 7

Apply the property of exponents from STEP_2 to add the exponents of the variables.
$x^{2} \cdot x^{7} = x^{2+7}$

STEP 8

Calculate the sum of the exponents for the variable $x$ .
$x^{2+7} = x^{9}$

STEP 9

Combine the results from STEP_6 and STEP_8 under a single cube root.
$\sqrt[3]{32} \cdot \sqrt[3]{x^{9}} = \sqrt[3]{32 x^{9}}$

STEP 10

Recognize that $32$ is $2^5$ and $x^9$ is $(x^3)^3$ . We can simplify the cube root of $x^9$ to $x^3$ because it is a perfect cube.
$\sqrt[3]{32 x^{9}} = \sqrt[3]{2^5} \cdot x^3$

STEP 11

Since $2^5$ is not a perfect cube, we can separate it into a product of a perfect cube and another factor to simplify the cube root.
$2^5 = 2^3 \cdot 2^2$

STEP 12

Recognize that $2^3$ is a perfect cube and its cube root is $2$ .
$\sqrt[3]{2^3} = 2$

STEP 13

Combine the result from STEP_12 with the remaining factor under the cube root.
$\sqrt[3]{2^5} \cdot x^3 = 2 \cdot x^3 \cdot \sqrt[3]{2^2}$

STEP 14

Simplify the expression by writing $2^2$ as $4$ .
$2 \cdot x^3 \cdot \sqrt[3]{2^2} = 2 \cdot x^3 \cdot \sqrt[3]{4}$

STEP 15

The simplest form of the product is therefore:
$2 x^{3} \cdot \sqrt[3]{4}$
This matches one of the given options.
The simplest form of the product is $2 x^{3} \cdot \sqrt[3]{4}$ .

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