Simplify cube root of x^2* square root of x
The problem is asking to perform algebraic manipulations on the expression given which involves both a cube root and a square root operation on variables. Specifically, you are required to simplify the expression that contains the cube root of x squared, which is written as $∛(x^2)$, multiplied by the square root of x, written as $√x$. Simplification in this context typically means to combine the roots and exponents by using the properties of exponents and radicals to express the entire expression as one radical if possible, and in the simplest form.
$\sqrt[3]{x^{2}} \cdot \sqrt{x}$
Answer
Solution:
Step 1
Begin by multiplying the expressions $\sqrt[3]{x^{2}}$ and $\sqrt{x}$ together.
Step 1.1
Convert the expressions to a common radical index, which is $6$ in this case.
Step 1.1.1
Transform $\sqrt[3]{x^{2}}$ into exponential form using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$, resulting in $x^{\frac{2}{3}}$.
Step 1.1.2
Express $x^{\frac{2}{3}}$ with a denominator of $6$, obtaining $x^{\frac{4}{6}}$.
Step 1.1.3
Convert $x^{\frac{4}{6}}$ back into radical form as $\sqrt[6]{x^{4}}$.
Step 1.1.4
Apply the same rule to $\sqrt{x}$ to rewrite it as $x^{\frac{1}{2}}$.
Step 1.1.5
Adjust the exponent of $x^{\frac{1}{2}}$ to have a denominator of $6$, yielding $x^{\frac{3}{6}}$.
Step 1.1.6
Rewrite $x^{\frac{3}{6}}$ as a sixth root, giving $\sqrt[6]{x^{3}}$.
Step 1.2
Combine the two sixth roots using the product rule for radicals to get $\sqrt[6]{x^{4} x^{3}}$.
Step 1.3
Simplify the expression inside the radical by adding the exponents.
Step 1.3.1
Apply the exponent rule $a^{m} a^{n} = a^{m + n}$ to combine the exponents inside the radical.
Step 1.3.2
Perform the addition of the exponents $4$ and $3$ to get $\sqrt[6]{x^{7}}$.
Step 2
Extract $x^{6}$ from $\sqrt[6]{x^{7}}$ by factoring it out.
Step 3
Finally, simplify the expression by taking $x^{6}$ out of the radical, which results in $x \sqrt[6]{x}$.
Knowledge Notes:
The solution involves several mathematical rules and properties:
Radical and Exponential Forms: The relationship between radicals and exponents is given by $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$. This allows us to convert between radical form and exponential form.
Common Indices: When dealing with multiple radicals, it's often useful to rewrite them with a common index to combine them more easily.
Product Rule for Radicals: The product rule states that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$. This allows us to combine radicals with the same index.
Exponent Rules: The power rule for exponents states that $a^{m} \cdot a^{n} = a^{m + n}$. This is used to combine like bases with exponents.
Simplifying Radicals: When an exponent inside a radical is a multiple of the index, we can simplify by taking out the base raised to the quotient of the exponent and index.
By applying these rules and properties, we can simplify complex radical expressions into a more manageable form.
